A NULLSTELLENSATZ FOR AMOEBAS

A NULLSTELLENSATZ FOR AMOEBAS 

KEVIN PURBHOO 

Abstract 

The amoeba of an affine algebraic variety V ⊂ (C ∗ ) r is the image of V under 

the map (z 1 ,...,z r ) ↦→ (log |z 1 |,...,log |z r |). We give a characterisation of the 

amoeba, based on the triangle inequality, which we call testing for lopsidedness. We 

show that if a point is outside the amoeba of V , there is an element of the defining ideal 

which witnesses this fact by being lopsided. This condition is necessary and sufficient 

for amoebas of arbitrary codimension as well as for compactifications of amoebas 

inside any toric variety. Our approach naturally leads to methods for approximating 

hypersurface amoebas and their spines by systems of linear inequalities. Finally, we 

remark that our main result can be seen as a precise analogue of a Nullstellensatz 

statement for tropical varieties. 

Contents 

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 

2. The case r = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 

3. The hypersurface case . . . . . . . . . . . . . . . . . . . . . . . . . . 417 

4. Approximating a hypersurface amoeba by linear inequalities . . . . . . 426 

5. More general amoebas . . . . . . . . . . . . . . . . . . . . . . . . . . 431 

6. Tropical varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 

Appendix. Details of calculations . . . . . . . . . . . . . . . . . . . . . . . 441 

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 

1. Introduction 

1.1. Statement of results 

Let V ⊂ (C ∗ ) r be an algebraic variety, defined by an ideal I ⊂ 

C[z 1 ,z −1 

1 ,...,z r,zr −1 ]. 

Definition 1.1 (Gel’fand, Kapranov, and Zelevinsky; see [GKZ]) 

The amoeba of V is defined to be the image of V under the map Log : (C ∗ ) r → R r 

DUKE MATHEMATICAL JOURNAL 

Vol. 141, No. 3, c○ 2008 DOI 10.1215/00127094-2007-001 

Received 24 July 2006. Revision received 5 April 2007. 

2000 Mathematics Subject Classification. Primary 14Q15; Secondary 14Q10, 14M25. 

Author’s research partially supported by Natural Science and Engineering Research Council of Canada. 

407

408 KEVIN PURBHOO 

defined at the point z = (z 1 ,...,z r ) by 

Log(z) = (log |z 1 |,...,log |z r |). 

We denote the amoeba of V by either A V or A I .IfV = Z f is a hypersurface, 

the zero locus of a single function f , we also use the notation A f . We refer the 

reader to Mikhalkin’s survey article [M] for a broad discussion of amoebas and their 

applications. 

In this article, we address the following fundamental question: given a point 

a ∈ R r and an ideal I ⊂ C[z 1 ,z −1 

1 ,...,z r,zr 

−1 ],whenisa ∈ A I ? This problem was 

previously studied by Theobald [T], who gave a practical answer for certain families 

of amoebas. Here we give a general answer to this question. We first consider the case 

where I =〈f 〉 is the ideal of a hypersurface. From this, we deduce a characterisation 

theorem for arbitrary ideals which is the analytic counterpart to a fundamental theorem 

for tropical varieties. 

Consider f ∈ C[z 1 ,z −1 

1 ,...,z r,zr 

−1 ], and consider a point a ∈ R r . Write f as a 

sum of monomials f (z) = m 1 (z) +···+m d (z).Definef {a} to be the list of positive 

real numbers 

f {a} := {∣ ∣ m1 

( 

Log −1 (a) )∣ ∣ ,..., 

∣ ∣md 

( 

Log −1 (a) )∣ ∣ } . 

Note that since the m i are monomials, this is well defined, even though Log is not 

injective. 

Definition 1.2 

We say that a list of positive numbers is lopsided if one of the numbers is greater than 

the sum of all the others. 

Equivalently, a list of numbers {b 1 ,...,b d } is not lopsided if it is possible to choose 

complex phases φ i (|φ i |=1), so that ∑ φ i b i = 0. This follows from the triangle 

inequality. We also define 

LA f := { a ∈ R r ∣ ∣ f {a} is not lopsided 

} 

. 

One can easily see that if a ∈ A f ,thenf {a} cannot be lopsided; in other words, 

LA f ⊃ A f .Indeed,iff (z) = 0, thenm 1 (z) +···+m d (z) = 0, soitisgivinga 

way to assign complex phases to the list {|m 1 (z)|,...,|m d (z)|} = f {Log(z)} such 

that the sum is zero. Thus one can think of LA f as a crude approximation to the 

amoeba A f . 

Example 1.3 

Suppose that f (z 1 ,z 2 ) = 1 + z 1 z 2 + z2 2, and let a ∈ R2 . For any complex 

phases φ 1 ,φ 2 , there exist (z 1 ,z 2 ) ∈ Log −1 (a) such that φ 1 |z 1 z 2 | = z 1 z 2 and

A NULLSTELLENSATZ FOR AMOEBAS 409 

φ 2 |z2 2|=z2 2 . Thus a ∈ A f 

A f = LA f . 

if and only if {1, |z 1 z 2 |, |z2 2 |} is nonlopsided; that is, 

In the above example, we have enough freedom to choose the phases of the monomials 

m i (z) for z ∈ Log −1 (a) so that LA f = A f . However, this works only because f 

has very few nonzero terms. In general, LA f can be quite different from A f (see 

Figure 1). Nevertheless, we show that for a suitable multiple of f , we can use this 

lopsidedness test to get very good approximations for A f . 

Let n be a positive integer. We consider the polynomials 

˜f n (z) = 

∏n−1 

k 1 =0 

These ˜f n are cyclic resultants 

··· 

∏n−1 

k r =0 

f (e 2πi k 1/n z 1 ,...,e 2πi k r /n z r ). 

˜f n (z) = Res ( Res(...Res(f (u 1 z 1 ,...,u r z r ),u n 1 − 1) ...,un r−1 − 1),un r − 1) 

and as such can be practically computed. When n = 2 k , this can be done reasonably 

efficiently as follows. Define polynomials h i by h 0 := f ,andlet 

h i (z 1 ,...,z 2 [i] ,...,z r):= h i−1 (z 1 ,...,z [i] ,...,z r ) h i−1 (z 1 ,...,−z [i] ,...,z r ), 

where [i] ∼ = i (mod r). Then ˜f n (z) = h kr (z1 n,...,zn r 

), and the recursion computes 

this in O(n 2(r2−r) )-time, which is proportional to the square of the number of terms of 

˜f n . 

Our main result for amoebas of hypersurfaces is roughly the following. The 

precise version is stated and proved in Section 3.2. 

THEOREM 1 (Rough version) 

As n →∞, the family LA ˜f n 

converges uniformly to A f . There exists an integer N 

such that to compute A f to within ε, it suffices to compute LA ˜f n 

for any n ≥ N. 

Moreover, N depends only on ε and the Newton polytope (or degree) of f and can be 

computed explicitly from these data. 

This leads us to the following characterisation of the amoeba of a general subvariety 

of (C ∗ ) r . 

THEOREM 2 

Let I ⊂ C[z 1 ,z −1 

1 ,...,z r,zr 

−1 ] be an ideal. A point a ∈ R r is in the amoeba A I if 

and only if for every g ∈ I, g{a} is not lopsided. 

Phrased another way, if a point a is outside the amoeba A I , a polynomial f ∈ I may 

witness this fact by being lopsided at a. Theorem 2 then states that there is always 

a witness. We actually show something slightly stronger in both Theorems 1 and 2.


Figure 1. The image on the left depicts LA f ⊃ A f , while the image on the right depicts 

SA f ⊃ A f . Here SA f is not homotopic to A f , and in general, 

LA f need not be either. 

We show that there is a witness f such that f {a} is superlopsided according to the 

following definition. 


Let d ′ ≥ d ≥ 2. We say that a list of positive numbers {b 1 ,...,b d+1 } is 

d ′ -superlopsided if there exists some i such that b i >d ′ b j for all j ≠ i. Ifd ′ = d, 

we simply say the list is superlopsided. 

As before, we also define 

SA f := { a ∈ R r ∣ ∣ f {a} is not superlopsided } . 

If a list of positive numbers is superlopsided, it is certainly lopsided; hence 

SA f ⊃ LA f ⊃ A f (see Figure 1). David Speyer [S] observed that each component 

of the complement of SA f is given by a system of linear inequalities, making it easier 

than LA f to compute explicitly. Hence Theorem 1 actually prescribes a method for 

approximating A f to within ε by systems of linear inequalities. Similar ideas lead 

to a method for approximating the spine of a hypersurface amoeba. We discuss these 

constructions in Section 4. 

The motivation for these results comes from tropical algebraic geometry, and 

from this viewpoint, lopsidedness (rather than superlopsidedness) is the more natural 

condition to consider. In tropical algebraic geometry, we work with the semiring 

R trop (⊙, ⊕). This is a semiring whose underlying set is R but whose operations are 

given by 

• a ⊙ b := a + b, 

• a ⊕ b := max(a,b). 

The operations ⊙ and ⊕ are known as tropical addition and tropical multiplication. One 

can easily check that they satisfy the usual commutative, associative, and distributive 

laws; however, there are no additive inverses.


A polynomial g ∈ R trop [x 1 ,...,x r ] is therefore a piecewise linear function on 

R r ;if 

g(x) = ⊕ 

c k1 ,...,k r 

⊙ x k 1 

1 ⊙···⊙xk r 

r , 

k 1 ,...,k r 

then, translated into the usual operations on R, 

g(x) = max{c k1 ,...,k r 

+ k 1 x 1 +···+k r x r }. 

The tropical variety associated to g is then defined to be the singular locus of this 

piecewise linear function. A tropical variety associated to a single polynomial g in 

this way is called a tropical hypersurface. 

Thus there is a simple Nullstellensatz ∗ for tropical hypersurfaces. A point x is 

outside the tropical variety of g if there is a single monomial term of g which is 

strictly larger than each of the others when evaluated at the point x. In terms of the 

tropical operations, this term is strictly greater than the tropical sum of the other terms 

(cf. Definition 1.2). 

More generally, the principal results in this article can be seen as an analytic 

analogue of a theorem for tropical varieties of arbitrary codimension (see [EKL,Theorem 

2.2.5], [SS, Theorem 2.1], [St, Theorem 9.17]), also known as nonarchimedean 

amoebas. We discuss this connection in Section 6. 

2. The case r = 1 

2.1. A heuristic argument 

The idea of the one-variable case is simple enough. Suppose that f (z) = ∏ d 

i=1 (α i −z), 

and for sake of argument, assume that the absolute values of the α i are all distinct, 

say, |α 1 | > ···> |α d | > 0.Then 

˜f n (z) = 

d∏ 

i=1 

(α n i 

− z n ) 

=±z nd ∓ (α n 1 + { ···})z n(d−1) 

± (α n 1 αn 2 + { ···})z n(d−2) 

∓ ···+ 

− (α n 1 ···αn d−1 + { ···})z n 

+ α n 1 ···αn d . 

∗ We use the term in the literal sense of being a statement about zeros; these results are not an analogue of 

Hilbert’s Nullstellensatz.


For n large, the terms { ···} are small in comparison with the other terms, and so 

this is approximately 

g n (z) =±(z d ) n ∓ (α 1 z d−1 ) n ± (α 1 α 2 z d−2 ) n ∓···−(α 1 ···α d−1 z) n + (α 1 ···α d ) n . 

Suppose that |α k+1 | < |z| < |α k |. Consider g n (z)/(α 1 ···α d−k z d−k ) n .Asn →∞, 

every term tends to zero except for the constant term, which is ±1. Thus, for n large, 

there is a single term in g n (z), and likewise in ˜f n , which is much bigger in absolute 

value than all the others. 

2.2. The one-variable lemmas 

We now formalise this heuristic argument in a way that is useful in proving Theorem 1. 

At the crux of the heuristic argument are the following three key facts about ˜f n . 

(1) It has no roots inside a certain annulus. (In the heuristic argument, the annulus 

is {z ∈ C ||α k+1 | < |z| < |α k |}.) 

(2) The only nonzero terms that appear are of the form cz nk . 

(3) The number of terms is not too large. (This approach fails if instead of ˜f n (z), 

we try to use ˜f n (z) D with D ≫ n.) 

To get a result that we can apply to the multivariable case, we need to be able to 

make a uniform statement about polynomials with these properties. This is precisely 

captured by the next two lemmas. By applying Lemma 2.2 directly to the family of 

functions ˜f n (z), one immediately obtains a complete proof of Theorem 1 in the r = 1 

case. 

LEMMA 2.1 

Let A ={z | β 0 < |z|


conclusion (2.1) remains valid. Here maxdeg(f ) and mindeg(f ) refer, respectively, 

to the largest and smallest exponents that appear in f (z). The notation in our proof 

assumes that f (z) is actually a polynomial. 

Proof 

We can write 

f (z) = 

d∏ 

(z n + α n i ), 

where |α 1 |≥···≥|α d |. We adopt the convention that α 0 = 0 and α d+1 =∞. 

Since f n (z) has no roots in A, wehave 

i=1 

α k+1 ≤ β 0


If l


In particular, for each l ≠ k, wehave 

|m ′ l (z 0)| ≤(1 + γ n ) d − 1. (2.5) 

However, we can do slightly better than this by noting that the smallest power of 

γ which appears on the right-hand side of inequality (2.3) isγ |k−l| . Thus, whereas 

(2.5) tells us that |m ′ l (z 0)| < ∑ ( d 

) 

w≥1 w γ nw , in fact, we have 

|m ′ l (z 0)| < ∑ ( d 

γ 

w) 

nw 

w≥|k−l| 

< ∑ 

w≥|k−l| 

(dγ n ) w 

. (2.6) 

w! 

(Although (2.3) is a better estimate than (2.6), the latter proves to be more useful to 

us.) 

For m k (z 0 ), we have the estimate 

∣ ( ∑ ) |m ′ k (z s 

0)| = 

1


Remark 2.1 

Note that we have actually determined which term is the special term m k .Thisis 

done in (2.2). If f n is a polynomial, then n(d − k) is the number of roots (counted 

with multiplicity) of f n inside the disc {|z| ≤β 0 }.Iff n is a Laurent polynomial, 

n(d − k) − mindeg(f n ) is the number of roots inside {0 < |z| ≤β 0 }. 

LEMMA 2.2 

As before, let A ={z | β 0 < |z|


γ → 1. The best general answer for the question is of the same form; that is, 

n log γ −1 ≤ (D 0 + D 1 ) log n + log (c 0 c 1 ). 

One can see this by performing the requisite analysis on the polynomials h n (z) = 

(z n + 1) c 0n D 0 

. As a general heuristic, the more closely the roots of f n are packed, the 

larger n has to be; thus the family of polynomials h n (z), where every root has as high 

a multiplicity as possible, is where we expect our worst-case behaviour to occur. 

Suppose we want n large enough to guarantee that h n {log |z 0 |} is (c 1 n D 1 )-superlopsided 

for |z 0 | < γ < 1. Write h n (z) = 1 + c 0 n D 0 zn + ···. We know that 

1 is the dominant term as n gets large (since 1 = lim n→∞ h n (z 0 )); thus we need 

(c 1 n D 1 )(c 0n D 0 zn 0 ) < 1 or, equivalently, 

n log γ −1 ≥ (D 0 + D 1 ) log n + log(c 0 c 1 ). 

If we only want to guarantee that f n {a} is lopsided for a ∈ K, we need n large 

enough so that 

∑ 

|m l (z 0 )|≤(1 + γ n ) d − 1 < 2 − (1 + γ n ) d ≤|m k (z 0 )| 

l≠k 

or, equivalently, (1 + γ n ) c 0n D 0 

≤ 3/2. This holds if we have 

( 

n log γ −1 c 

) 

0 

≥ D 0 log n + log . 

log 3/2 

So n needs to be only about half as big to guarantee that f n {a} is lopsided as it does 

to guarantee that f n {a} is superlopsided. Again, we can see that this is fairly close to 

the best answer by considering (z n + 1) c 0n D 0 

. 

3. The hypersurface case 

3.1. Preliminaries 

In this section, we prove our main theorem characterising the amoeba of a hypersurface. 

If f (z) ∈ C[z 1 ,z −1 

1 ,...,z r,z −1 ], we consider the Laurent polynomials 

˜f n (z) = 

∏n−1 

k 1 =0 

r 

··· 

∏n−1 

k r =0 

f (e 2πi k 1/n z 1 ,...,e 2πi k r /n z r ). 

Theorem 1 states that for ε>0 and for a point a ∈ R r in the complement of the 

amoeba A f whose distance from A f is at least ε, we can choose n large enough so 

that ˜f n {a} is superlopsided. Moreover, the theorem gives an upper bound on how large 

n needs to be, based only on ε and the Newton polytope of f .


The idea behind the proof of Theorem 1 is to look at the family of ˜f n (z) and 

interpret this as a function of a single variable z i . At the point ζ = (ζ 1 ,...,ζ r ) ∈ C r , 

we define 

˜f i,ζ 

n (z) := ˜f n (ζ 1 ,...,ζ i−1 ,z,ζ i+1 ,...,ζ n ). 

We apply Lemma 2.2 to these and find a single dominant term in this polynomial of 

one variable. Then by an averaging argument, we show that this implies that ˜f n has a 

single dominant term. 

First, however, we need a few simple observations. 

PROPOSITION 3.1 

We have A f = A ˜f n 

. 

Proof 

The cyclic resultant ˜f n (z) is a product of terms g u1 ,...,u r 

(z) = f (u 1 z 1 ,...,u r z r ), where 

u n i = 1.Since|u i |=1, A gu1 ,...,ur 

= A f , and so A ˜f n 

= ⋃ A gu1 ,...,ur 

= A f . 

We also need to know some information about the number and degree of the terms 

which appear in ˜f n . First, note the following important fact. 


The only monomials that appear in ˜f n are of the form cz nk 1 

1 ···z nk r 

r 

. In particular, the 

only terms appearing in ˜f 

n i,ζ(z) 

are of the form cznk . 

Proof 

Let C n denote the cyclic group of roots of z n −1.Now, ˜f n is manifestly invariant under 

the group action of (C n ) r acting on C[z 1 ,z −1 

1 ,...,z r,zr −1 ] by (u 1 ,...,u n ) · g(z) = 

g(u 1 z 1 ,...,u n z n ). Thus each monomial of ˜f n must be invariant under this action. 

The only monomials with this property are of the form cz nk 1 

1 ···z nk r 

r 

. 

The statement about ˜f 

n i,ζ (z) follows immediately. 

Recall that if g ∈ C[z 1 ,z −1 

1 ,...,z r,zr 

−1 ], then its Newton polytope, denoted (g),is 

the subset of R r defined as the convex hull of the exponent vectors of the monomials 

which appear in g. 

For any polytope , letd() be any upper bound on (#{Z r ∩ m})/m r .In 

general, it is not easy to find a tight upper bound for this number. If one can compute 

the Ehrhart polynomial of explicitly, then an easy upper bound is the sum of the 

positive coefficients. Otherwise, it is possible to bound the coefficients of the Ehrhart 

polynomial in terms of the volume of (see [BM]). Using these estimates, for each r 

one can compute constants A and B such that (#{Z r ∩ m})/m r


Clearly, we have ( ˜f n ) = n r (f ). This gives us an upper bound on the number 

of terms that ˜f n can have. 


Let d = d((f )). Then ˜f n has at most dn r2 −r terms. 

Proof 

By Proposition 3.2, the number of terms in ˜f n is at most the number of integral points 

in (1/n)( ˜f n ) = n r−1 (f ). This is less than or equal to dn r2−r . 

 

Finally, we need to know something about maxdeg( ˜f i,ζ 

n 

c i (f ):= max x i 

( 

(f ) 

) 

− min xi 

( 

(f ) 

) 

, 

where x i denotes the ith coordinate function on R r . 

) − mindeg( ˜f 

n 

i,ζ).Let 


We have maxdeg( ˜f i,ζ 

n 

Proof 

We have 

maxdeg( ˜f i,ζ 

n 

) − mindeg( ˜f i,ζ 

n ) = c i(f )n r . 

) − mindeg( ˜f i,ζ 

n 

) = max x ( 

i ( ˜f n ) ) ( 

− min x i ( ˜f n ) ) 

( 

= n r max x i ( ˜f n ) ) ( 

− n r min x i ( ˜f n ) ) 

= d i n r . 

3.2. Proof of Theorem 1 

Armed with these facts and Lemmas 2.1 and 2.2, we are now in a position to precisely 

state and prove our main result for amoebas of hypersurfaces. 

THEOREM 1 

Let ε>0. Suppose that a = (a 1 ,...,a r ) ∈ R r \ A f is a point in the amoeba 

complement whose distance from A f is at least ε. Letd = d((f )), and let c = 

max{c i (f ) | 1 ≤ i ≤ r}. 

(1) If n is large enough so that 

then ˜f n {a} is lopsided. 

nε ≥ (r − 1) log n + log ( (r + 3)2 r+1 c ) , (3.1)


(2) If n is large enough so that 

( (16 ) ) 

nε ≥ (r 2 − 1) log n + log cd , (3.2) 

3 

then ˜f n {a} is superlopsided. (In fact, it is (dn r2 −r )-superlopsided.) 

The key to reducing to the one-variable case is the following basic result from complex 

analysis. 

LEMMA 3.5 

Let f (z) be a Laurent polynomial, and write f (z) = ∑ −→ m−→ j j 

(z), wherem−→ j 

(z) = 

m j1 ,...,j r 

(z) = b j1 ,...,j r 

z j 1 

1 ···zj r 

r . Suppose that for all ζ ∈ Log−1 (a), we have |f (ζ)| ≤ 

M. Then for each −→ l, |m−→ l 

(ζ)| ≤M. 

Proof 

We integrate the equations M ≥|f (ζ)| over the set Log −1 (a 1 ,...,a r ): 

M ≥ 1 

(2π) r ∫ 2π 

θ 1 =0 

1 

≥ ∣ 

(2πi) 

∫|z r 1 |=1 

∫ 2π 

··· ∣ ∑ m−→ j 

(e a 1+iθ 2 

,...,e a r +iθ r ) ∣ dθ 1 ···dθ r 

−→ j 

θ r =0 

=|m−→ l 

(e a 1 

,...,e a r 

)| 

∫ 

∑ 

··· 

|z r |=1 

m−→ j 

(e a 1 z 1,...,e a r z r) 

−→ z l 1 

1 ···z l r 

r 

j 

dz 1 

z 1 ···dz 1 

z 1 

∣ ∣∣ 

=|m−→ l 

(ζ)|. 

 

Proof of Theorem 1 

Let γ = e −ε ,andletA i ={z | γe a i 

< |z| ε. But since z 0 ∈ A i , 

‖Log(ζ ′ ) − a‖ =|log(z 0 ) − a i |


fixed i, it must be the same monomial term that dominates in each ˜f 

n 

i,ζ, 

independent 

of the choice of ζ. Let this be the z nk i 

-term, and let −→ k = (k 1 ,...,k r ). 

Write 

˜f n (z) = ∑ −→ j 

m−→ j 

(z), 

where m−→ j 

(z) is the monomial b−→ j 

z nj 1 

1 ···z nj r 

r 

. 

Let M =|m−→ k 

(ζ)|. Note that this does not depend on the particular choice of ζ. 

Let 

µ = max { |m−→ l 

(ζ)| ∣ ∣ −→ l ≠ −→ k } . 

We wish to show that µ


We now prove statement (1). Although the approach is essentially the same, it is 

slightly more difficult and hence requires some additional lemmas (Calculations A.2 

and A.3, which are found in the appendix). The reason for this is that we cannot get 

these bounds by appealing directly to Lemma 2.2. Instead, we use Lemma 2.1,which 

gives better estimates for the coefficients of ˜f 

n i,ζ. 

Write ˜f i,ζ 

maxdeg ˜f 

n 

i,ζ 

that 

n (z) = ∑ j mi j (z), where mi j (z) = b jz nj . Then for each i, 

(z) − mindeg ˜f 

n 

i,ζ(z) 

≤ cnr . So by Lemma 2.1, there is some k i such 

|m i l (ζ)| 

∑ 

|m i k i 

(ζ)| < w≥|k i −l| (cnr−1 γ n ) w /w! 

, 

2 − e cnr−1 γ n 

and in fact, it is the same k i for all choices of ζ. 

As before, let M =|m−→ k 

(ζ)|, andletσ = ∑ −→ j ≠ 

−→ k 

|m−→ j 

(ζ)|. Wehaveforanyζ 

that |m i k i 

(ζ)|


and by Calculation A.3 (see the appendix), this becomes 

σ 

( e 

(r+2)cn r−1 γ n 

M + σ < − 1 

) 

2r . (3.5) 

2 − e cnr−1 γ n 

Assume now that (3.1) holds. By Calculation A.2 (see the appendix), n is large 

enough so that the right-hand side is less than 1/2. Thus we have σdcan be used in place of d in Theorem 1. 

Thus we see that it suffices to take n so that 

( (16 ) ) 

nε ≥ (r 2 − 1) log n + log cα . 

3


3.4. Accuracy of bounds 

Just as was the case in Lemma 2.2, the bounds on n given in Theorem 1 are not quite 

optimal; there are a number of places in which the inequalities can obviously be made 

tighter. However, as ε → 0, the bounds are at least asymptotically correct. 

To see this for the superlopsided case, we can consider the example 

f (z) = (1 − z 1 ) D1 ···(1 − z r ) D r 

. 

The amoeba A f 

compute 

is the union of all coordinate hyperplanes in R r . We can easily 

˜f n (z) = (1 − z n 1 )D 1n r−1 ···(1 − z n r )D r n r−1 

= 1 − D 1 n r−1 z n 1 −···−D rn r−1 z n r +···. 

If our point is a = (a 1 ,...,a r ), with each −ε D i (D 1 ···D r )n r2 −r 

nε > (r 2 − 1) log n + log ( (D 1 ···D r )max{D i } ) . 

In contrast, (3.2) says that in this example, we should take n so that 

( (16 ) 

) 

nε > (r 2 − 1) log n + log (D 1 + 1) ···(D r + 1) max{D i } . 

3 

Our bound (3.1) for lopsidedness appears to be slightly less satisfactory. In the 

above example, to guarantee lopsidedness one needs n large enough so that 

This holds when 

(1 − z n 1 )D 1n r−1 ···(1 − z n r )D r n r−1 − 1 < 1. 

(ea 1 n + 1) D 1n r−1 ···(e a r n + 1) D r n r−1 < 2 

⇔D 1 n r−1 log(1 + e a 1n ) +···+D r n r−1 log(1 + e a r n ) < log 2.


Noting that a i > −ε, and approximating log(1 + x) ∼ x, this condition becomes 

(D 1 +···+D r )n r−1 e −εn < log 2 

( D1 +···+D r 

⇔ nε > (r − 1) log n + log 

log 2 

If we take D 1 =···=D r = D, then this simplifies to 

( rD 

) 

nε > (r − 1) log n + log . 

log 2 

In contrast, Theorem 1 tells us that it is sufficient to take n so that 

nε > (r − 1) log n + log ( (r + 3)D ) + (r + 1) log 2. 

Again, this shows that the bounds in Theorem 1 are asymptotically correct, at least 

for any fixed r. We suspect, however, that the correct general answer does not have 

this last term, or any term which is linear in r. 

3.5. Other cyclic resultants 

Instead of the family ˜f n , one may wish to consider a more general family of cyclic 

resultants. Let n 1 ,...,n r be positive integers, and consider 

˜f n1 ,...,n r 

(z) = 

n∏ 

1 −1 

k 1 =0 

n∏ 

r −1 

··· 

k r =0 

) 

. 

f (e 2πi k 1/n 1 

z 1 ,...,e 2πi k r /n r 

z r ). 

Unfortunately, it is not true that the family SA ˜f n1 

converges uniformly to A 

,...,nr 

f 

as n 1 ,...,n r →∞. Trouble occurs if some of the n i are significantly larger than 

others. For example, consider the amoeba of f (z 1 ,z 2 ) = (1 − z 1 )(1 − z 2 ) at a point 

(a 1 ,a 2 ) ∈ R 2 with a 1 < 0, a 2 < 0. Then 

˜f n1 ,n 2 

(z 1 ,z 2 ) = (1 − z n 1 

1 )n 2 

(1 − z n 2 

2 )n 1 

= 1 − n 2 z n 1 

1 − n 1z n 2 

2 +···. 

If n 2 ∼ e −a 1n 1 

, then the first two terms above will have the same order of magnitude. 

Thus ˜f n1 ,n 2 

{(a 1 ,a 2 )} is not superlopsided, even if n 1 (and hence n 2 ) are large. It is not 

even lopsided. 

However, if we restrict ourselves to the situation in which each n i is bounded by 

some polynomial in each of the other n j , then a statement analogous to Theorem 1 

is true. For example, we can let n i be any polynomial function of a single parameter 

n. We do not compute explicit bounds for approximating the amoeba to within ε in


this more general situation; however, the answer depends on these polynomials. It is 

certainly still true that SA ˜f n1 

converges uniformly to A 

,...,nr 

f , as this argument really 

only depends on the fact that degrees of ˜f n1 ,...,n r 

are growing only polynomially, while 

the terms are becoming suitably sparse. 

4. Approximating a hypersurface amoeba by linear inequalities 

4.1. Locating the dominant term 

Theorem 1 tells us that for n sufficiently large, one term of ˜f n dominates, but it does 

not specify which one. The answer depends on in which component of the amoeba 

complement our point a lies. Since ˜f n {a} varies continuously with a, it depends only 

on the component of the amoeba complement. 

Now, the number of components is relatively small compared to the number of 

terms of ˜f n . There is a natural injective map 

ind : components of R r \A f ↩→ (f ) ∩ Z r 

(cf. [FPT, Definition 2.1]). This is called the index of the component; a complete 

definition is given below. So only a few of the terms of ˜f n can possibly be dominant 

terms. Fortunately, it is relatively simple to determine which these are. The Newton 

polytope of ˜f n is n r (f ), and candidates for dominant term are, in fact, what one 

expects them to be; namely, they are the images of the integral points of (f ) under 

this scaling. 


Let a ∈ R r \A f , and let ind(a) = −→ k = (k 1 ,...,k r ) be the corresponding point 

in (f ). If ˜f n {a} is lopsided, then the term of ˜f n (z) which dominates has exponent 

vector n r−→ k (i.e., it is the (z nr k 1 

1 ···z nr k r 

)-term). 

r 

In order to make complete sense of the statement, we need to know a definition of 

the index −→ k . There are a number of equivalent definitions, but the simplest for our 

purposes is the following. 

Let ζ ∈ Log −1 (a). For each i ∈{1,...,r}, consider the polynomial, 

f i,ζ (z) = f (ζ 1 ,...,ζ i−1 ,z,ζ i+1 ,...,ζ r ). 

If f is a polynomial, then k i is the number of roots (with multiplicity) of f i,ζ inside 

the open disc {|z|


ζ, this number is independent of ζ.Iff is a Laurent polynomial, then 

k i = #roots of f i,ζ (z) inside { 0 < |z|


is the candidate for the dominant term in this component, and m−→ j 

(z) = b−→ j 

z j 1 

1 ···zj r 

r 

are the other monomials. 

The corresponding component of R r \SA ˜f n 

is the set 

Log ({ z ∣ ∣ |M −→ k 

(z)| >D|m−→ j 

(z)|, ∀ −→ j }) , 

where D + 1 is the number terms in ˜f n . Equivalently, this is the set of x ∈ R r such 

that 

log|B−→ k 

|+n r k 1 x 1 +···+n r k r x r > log D + log|b−→ j 

|+j 1 x 1 +···+j r x r (4.1) 

for all −→ j . This is a system of linear inequalities in the variable x, so the solutions 

to these equations are a convex polyhedron that approximates the component of the 

amoeba to within ε. If there is no component of the R r \A f corresponding to −→ k ,then 

this system of equations has no solutions. Conversely, if this system of inequalities 

has no solutions, then this component of the amoeba (if it exists) is not large enough 

to contain a ball of radius ε. 

Thus we can realise any component of the R r \A f as an increasing union of 

convex polyhedra. This gives an independent proof of the basic fact (see [FPT]) that 

the components of the R r \A f are convex. We must admit, however, that there are 

simpler proofs of this fact. 

Note that in Theorem 1, we actually show that ˜f n is (dn r2−r )-superlopsided. Thus 

we can, in fact, take D = dn r2−r in (4.1), and the set of solutions to this system of 

inequalities still approximates the component of R r \A f to within ε. 

In practice, it rapidly becomes impractical to get arbitrarily good approximations 

to the amoeba by linear inequalities in this way, particularly for r > 2, since the 

number of inequalities is O(n r2−r ).Forr = 2, this is more manageable, though for 

purposes of simply drawing the amoeba, Theobald’s numerical method for drawing 

planar amoebas (see [T]) is probably faster. It is therefore natural to wonder whether 

some smaller subset of these inequalities can suffice. Although the answer is yes, 

it is unfortunately not easy to give an a priori answer as to which inequalities are 

needed. As n →∞, the terms m−→ j 

that are “near” to M−→ k 

become more relevant 

than the terms that are farther; however, this is only heuristic, and moreover, since we 

are approximating a piecewise smooth region by polyhedra, the number of relevant 

inequalities also approaches infinity. On the other hand, one practical use of Theorem 1 

is to find components of R r \A f , and here the heuristic that nearby terms are the 

most relevant can be helpful. One can first look for a value of x = Log(z) such that 

|M−→ k 

(z)| ≫|m−→ j 

(z)| for nearby terms m−→ j 

, and if one exists, check that x satisfies 

all inequalities (4.1). The efficiency of such an algorithm is commensurate with the 

computation of ˜f n .


4.3. Approximating the spine 

One of the primary tools for studying amoebas has been the Ronkin function N f , 

defined in [R]. For f ∈ C[z 1 ,z −1 

1 ,...,z r,zr 

−1 ], N f is defined to be the pushforward 

of log|f | under the map Log: 

N f (x) := 1 log|f (z 1 ,...,z r )| dz 1 ···dz r 

. 

(2πi) 

∫Log r −1 (x) z 1 ···z r 

Ronkin shows in [R] thatN f is a convex function, and it is affine-linear precisely on 

the components of R r \A f . When restricted to a single component of E of R r \A f , 

∇N f = ind(E). 

Passare and Rullgård [PR] use this function to define the spine of the amoeba as 

follows. For each component C of R r \A f , extend the locally affine-linear function 

of N f | E to an affine-linear function N E on all of R r .Let 

N ∞ f 

{ 

(x) = max NE (x) } . 

E 

This is a convex piecewise linear function on R r , superscribing N f .Thespine of the 

amoeba A f is defined to be the set of points where Nf 

∞ is not differentiable and is 

denoted S f . 

The spine of the amoeba S f is a strong deformation retract of A f (see [PR], [Ru]). 

Also, note that S f is actually a tropical hypersurface, as defined in the introduction; 

that is, it is the singular locus of the maximum of a finite set of linear functions, where 

the gradient of each linear function is a lattice vector. 

Now, observe that 

1 

n r log| ˜f n (z)| = 1 n r 

n∑ 

··· 

k 1 =1 

n∑ 

log|f (e 2πi k1/n z 1 ,...,e 2πi k r /n z r )| 

k r =1 

can be thought of as a Riemann sum for N f . In particular, we may expect 

(1/n r ) log| ˜f n (z)| to converge pointwise to N f (Log(z)). This is certainly true, provided 

that log| ˜f n (z)| is bounded on Log −1 (x), which is the case when x ∈ R r \A f . 

Suppose that x = Log(z) is in the component of R r \A f of index −→ k ∈ (f ). 

Assume that x has distance at least δ from the amoeba, where δ>0 is fixed. For any 

ε>0, we can find n sufficiently large so that 

˜f n (z) = M−→ k 

(z) + ∑ −→ j 

m−→ j 

(z), 

where each m−→ j 

is relatively small; that is, 

∑ 

|m −→ j 

(z)|


(see Corollary 3.7). Thus we have 

log|M−→ k 

(z)|+log(1 − ε) ≤ log| ˜f n (z)| ≤log|M−→ k 

(z)|+log(1 + ε). 

Thus we see that as n →∞, the values of (1/n r ) log| ˜f n (z)|, 

1 

n r log|M −→ k 

(z)| = 1 n r log|B −→ k 

|+k 1 x 1 +···+k r x r , 

and N f (x) = N E (x) = c−→ k 

+ k 1 x 1 +···+k r x r all converge on N f (x). 

We can use this fact to obtain good approximations for the spine of the amoeba. 

For each n, we consider the function M ∞ : R r → R given by 

M ∞ (x) := max log|M−→ k 

(z)|, (4.2) 

where the maximum is taken over all components of R r \A f . This is a piecewise 

linear function. We define the approximate spine of the amoeba LS f,n to be the set 

of points where M ∞ (x) is not smooth. Equivalently, LS f,n is the set of points where 

the maximum in equation (4.2) is attained by two distinct values of −→ k . 


We have the following relationships: 

(1) LS f,n ⊂ LA ˜f n 

; 

(2) lim n→∞ LS f,n = S f . 

Proof 

Statement (1) is true because on the component of R r \LA ˜f n 

of index −→ k , |M−→ k 

(z)| > 

|M−→ l 

(z)| for any other −→ l ∈ (f ). Thus the maximum value in equation (4.2) cannot 

be attained by two distinct −→ k if x ∈ R r \LA ˜f n 

. 

Statement (2) follows from the fact that (1/n r ) log|M−→ k 

(z)|−N f (x) is a constant 

function and is less than ε for n large. Let E 1 and E 2 be components of R r \A f of index 

−→ k 1 and −→ k 2 , respectively. Consider the hyperplane H ⊂ R r , where log|M−→ k 1 

(z)| and 

log|M−→ k 2 

(z)| coincide, and the hyperplane H ′ , where N E1 (x) and N E2 (x) coincide. The 

two hyperplanes H and H ′ are parallel, and their distance apart is most εK, where 

K is some constant depending only on −→ k 1 and −→ k 2 . As there are only finitely many 

−→ k ∈ (f ) ∩ Z r , these distances can be made uniformly small. 

 

For practical reasons, we may use an alternate definition of M ∞ , in which one takes 

the maximum in equation (4.2) over only those components that appear in R r \SA ˜f n 

. 

If we do, statement (2) is still true, and statement (1) is true for large n; for small n, 

we must settle for saying that LS f,n ⊂ SA ˜f n 

.


One may hope to be able to simplify this construction by taking the maximum in 

equation (4.2) over all −→ k ∈ (f ) ∩ Z, rather than just those that actually correspond 

to components. It appears, however, that this does not give the same answer. With this 

alternate definition of M ∞ ,theapproximate spine has false chambers for all n; that 

is, the complement of this approximate spine has components that do not correspond 

to components of R r \A f . We may still hope that these false chambers shrink to zero 

volume as n gets large. Unfortunately, experimental evidence suggests that the limit of 

these false chambers, as n →∞, can sometimes contain a ball of positive radius, and 

so this method does not produce a good approximation of the spine. An interesting 

open question is whether the limiting behaviour of these false chambers is somehow 

captured by the Ronkin function. 

5. More general amoebas 

5.1. Amoebas of higher codimension varieties in (C ∗ ) r 

The higher-codimension statement (Theorem 2) follows fairly quickly from the hypersurface 

statement. Let V ⊂ C r be a variety that is the zero locus of an ideal 

I =〈f 1 ,...,f k 〉⊂C[z 1 ,z −1 

1 ,...,z r,zr −1 ]. 


For every a ∈ R r , there exists f a ∈ I such that 

Proof 

For any Laurent polynomial 

Z fa ∩ Log −1 (a) = V ∩ Log −1 (a). 

g(z) = ∑ b−→ j 

z j 1 

1 ···zj r 

r 

∈ C[z 1 ,z −1 

1 ,...,z r,z −1 

r 

], 

−→ j 

let ḡ denote its complex conjugate 

ḡ(z) = ∑ −→ j 

¯b−→ j 

z j 1 

1 ···zj r 

r . 

We define f a to be 

f a (z) := 

k∑ 

i=1 

f i (z 1 ,...,z r ) ¯f i (e 2a 1 

z −1 

1 ,...,e2a r 

z −1 

r 

).


Clearly, f a is a Laurent polynomial and is in I. Moreover, if we restrict z to Log −1 (a), 

then z i ¯z i = e 2a i 

,so 

f a (z) = 

= 

= 

k∑ 

f i (z 1 ,...,z r ) ¯f i (¯z 1 ,...,¯z r ) 

i=1 

k∑ 

f i (z 1 ,...,z r )f i (z 1 ,...,z r ) 

i=1 

k∑ 

|f i (z)| 2 . 

i=1 

Thus f a (z) = 0 if and only if f i (z) = 0 for all i. 

 

This result is also true for ideals in C[z 1 ,...,z r ]: one can find a suitable monomial 

m(z) such that 

m(z 1 ,...,z r ) ¯f i (e 2a 1 

z −1 

1 ,...,e2a r 

z −1 

r 

) 

is a polynomial for all i, and a similar argument holds if 

f a (z) = 

k∑ 

i=1 

f i (z 1 ,...,z r ) ( m(z 1 ,...,z r ) ¯f i (e 2a 1 

z −1 

1 ,...,e2a r 

z −1 

r 

) ) . 

As an immediate consequence of Proposition 5.1, we have the following. 

COROLLARY 5.2 

For any ideal I ⊂ C[z 1 ,z −1 

1 ,...,z r,z −1 

r 

], 

A I = ⋂ f ∈I 

A f . 

It is now a simple task to prove our second main result. 

THEOREM 2 

Let I ⊂ C[z 1 ,z −1 

1 ,...,z r,zr 

−1 ] be an ideal. A point a ∈ R r is in the amoeba A I if 

and only if g{a} is not (super)lopsided for every g ∈ I. 

Proof 

If a ∈ A I ,thenf {a} cannot be lopsided for any f ∈ I since a ∈ A f for every 

f ∈ I. On the other hand, suppose that a /∈ A I . Then by Proposition 5.1, ifwetake


g = f a ∈ I, thena /∈ A g . By Theorem 1, ifn is sufficiently large, then ˜g n {a} is 

(super)lopsided, and ˜g n ∈ I. 

 

Remark 5.1 

In summary, we have three coincident sets, any of which can be used to define the 

amoeba A V of a variety V = V (I): 

(1) A V = Log(V ); 

(2) A V = ⋂ f ∈I A f ;and 

(3) A V ={a ∈ R r | f {a} is not lopsided for all f ∈ I}. 

In Section 6, we see that this is precisely analogous to a theorem for tropical algebraic 

varieties. 

If a point a is in R r \A I , the proof of Theorem 2 also tells us where to look for a 

witness to this fact: namely, we should look at ˜f an {a} for all n. For some sufficiently 

large n, this list is lopsided. 

One unfortunate misfeature of this proof is that it requires us to use a different 

g for every point a ∈ R r \A I . Thus this statement is purely local. It does not give 

any clues as to how to produce a global uniform approximation to A I .However,in 

general, we cannot expect there to be any finite set of elements g i ∈ I such that if 

a /∈ A I , then some ˜g i n is lopsided for n sufficiently large. If it were so, this would 

imply that A I is always an intersection of finitely many hypersurface amoebas, and 

this is certainly not true for dimensional reasons if dim V


More concretely, if we identify T ′ with (S 1 ) r′ , we can write 

χ(µ 1 ,...,µ r ′) = 

( r ′ 

∏ 

i=1 

∏r ′ 

) 

j ,..., µ A ir 

j , 

where A ij are the integer entries of a matrix A—the matrix representation of ˆχ—and 

µ A i1 

( r∑ 

Log ′ (z) = A Log(z) = A 1j log|z j |,..., 

j=1 

i=1 

r∑ 

j=1 

) 

A r ′ j log|z j |, . 

We can also take a matrix A with integer entries as our starting point and construct 

T ′ , χ, and the map Log ′ so that (5.1) holds. 

Let I ⊂ C[z 1 ,z −1 

1 ,...,z r,zr −1 ] denote the ideal of V. Let Ṽ ⊂ (C ∗ ) r+r′ denote 

the variety of the ideal 

Ĩ = I + J ⊂ C[z 1 ,z −1 

1 ,...,z r,z −1 

r 

,w 1 ,w −1 

1 ,...,w r ′,w−1 r ], ′ 

where J = 〈 w i − ∏ r 

〉 

j=1 zA ij 

j . Now, consider the projection of Ṽ onto the w-coordinates 

(C ∗ ) r′ . The image of Ṽ under this projection is a variety V ′ . Standard techniques of 

elimination theory allow us to compute its ideal I ′ (see, e.g., [CLO]). 


We have the following relationships: Log ′ (V ) = A(A V ) = A V ′. 

Proof 

A point in Ṽ is simply a pair (z, w), where z ∈ V and w i = ∏ r 

A Ṽ = { (x, y) ∈ R r+r′ ∣ ∣ y = Ax, x ∈ AV 

} 

. 

Projecting onto the w-coordinates, we obtain 

A V ′ = { ∣ 

y ∈ R r′ (x, y) ∈ A Ṽ for some x } 

={Ax | x ∈ A V } 

= A Log(V ) 

j=1 zA ij 

j 

. Thus we have 

= Log ′ (V ). 

It is interesting to note that this construction is closely related to the cyclic resultants 

used in the proof of Theorem 1. Suppose that V = Z f is a hypersurface, and suppose 

that χ : T ′ = T → T is the map χ(t) = t n . In this case, A = nI is a multiple of 

the identity matrix, and the variety V ′ is the zero locus of the function ˜f n . Intuitively, 

we should think that the linear transformation is zooming in on the amoeba A;aswe


zoom in, Theorem 1 tells us that we see more and more detail in the approximations 

LA and SA. 

5.3. Compactified amoebas 

The most natural generalisation of amoebas in the compact setting is to subvarieties of 

projective toric varieties. Each projective toric variety is a compactification of (C ∗ ) r 

with an (S 1 ) r -action which extends the (S 1 ) r -action on (C ∗ ) r . It also carries a natural 

symplectic form ω, for which the (S 1 ) r -action is Hamiltonian. We may therefore use 

the moment map for this Hamiltonian action to replace the map Log. 

Our goal in this section is to give a concrete description of this more general setting 

and observe that our results still hold. This follows fairly easily from the noncompact 

case. Our construction of toric varieties and their moment maps roughly follows a 

combination of [F]and[A]. 

Let ⊂ R r be an r-dimensional lattice polytope; that is, the vertices of have 

integral coordinates. To every such , we can associate the following data: 

(1) a set of lattice points A = ∩ Z r ; 

(2) a semigroup ring C[A]; ifA = { −→ k 1 ,..., −→ k d }, this is defined to be the 

quotient ring 

C[s −→ k 1 

,...,s −→ k d 

]/J, 

where each s −→ k 1 

has degree 1 and J is generated by all (homogeneous) relations 

of the form 

whenever 

s −→ k i1 ···s 

−→ k ip − s 

−→ k j1 ···s 

−→ k jp = 0 

−→ k i1 + ··· + −→ k ip = −→ k j1 + ··· + −→ k jp ; 

note that C[A] carries an action of the complex torus T = (C ∗ ) r ,givenby 

(λ 1 ,...,λ r ) · s −→k = λ k 1 

1 ···λk r 

r s−→k ; 

(3) a toric variety X = Proj(C[A]); 

(4) a projective embedding φ : X↩→ P d−1 = Proj(C[t 1 ,...,t d ]), induced by the 

map on rings C[t 1 ,...,t d ] → C[A] given by t i ↦→ s −→ k i 

; 

(5) a symplectic form ω = φ ∗ (ω P d−1), where ω P d−1 is the Fubini-Study symplectic 

form on P d−1 ;


(6) a moment map µ for the (S 1 ) r -action on (X, ω); we can, in fact, write down 

the moment map µ explicitly: 

µ(x) = 

1 

∑ d 

i=1 |s−→ k i(x)| 2 

d∑ 

|s −→ k i 

(x)| 2−→ k i ; 

to evaluate the right-hand side, we must choose a lifting of x to ˜X = 

Spec(C[A]); however, since this expression is homogeneous of degree zero 

in the s −→ k i 

, it is, in fact, well defined. 

It is well known that µ(X) = and that if Y is any other projective toric variety 

with µ Y (Y ) = , thenY ∼ = X as toric varieties. 

Let I ⊂ C[A] be a homogeneous ideal, and let V = Proj(C[A]/I) be its variety 

inside X. 

Definition 5.2 (Gel’fand, Kapranov, and Zelevinsky; see [GKZ]) 

The compactified amoeba of V is µ(V ) ⊂ . We denote the compactified amoeba of 

V by either A V or A I (or by A f if I =〈f 〉 is principal). 

Let f ∈ C[A] be a homogeneous polynomial of degree w. We can again decompose 

f as a sum of monomials; that is, write f = ∑ l 

i=1 m i, where each m i is a T-weight 

vector in C[A]. Each of these m i is a well-defined function on ˜X. Leta ∈ . We 

define f {a} := {|m 1 (ã)|,...,|m l (ã)|}, where ã is any preimage of a in the composite 

map ˜X → X → . Of course, f {a} depends on the choice of lifting under ˜X → X, 

but only up to rescaling. Thus the notions of f {a} being lopsided or superlopsided are 

still well defined. We define LA f and SA f to be the set of points a ∈ such that 

f {a} is nonlopsided and nonsuperlopsided, respectively. 

Let V ◦ denote the intersection of V with the open dense subset of X on which 

T acts locally freely. (A finite quotient of T acts freely.) We can identify this open 

dense subset with (C ∗ ) r and therefore consider A V ◦. As both Log and µ| (C ∗ ) r are 

submersions with fibres (S 1 ) r , it follows that A V ◦ is diffeomorphic to A V ∩ ◦ , 

where ◦ denotes the interior of . Letψ : ◦ → R r denote this diffeomorphism. 

Moreover, any face ′ of corresponds to a toric subvariety X ′ ⊂ X. And 

A V ∩ ′ = A V ∩X ′ (see [GKZ]). 

Thus, for every point a ∈ , we can determine whether a is in the compactified 

amoeba A V as follows. First, we determine the face ′ ⊂ for which a ∈ ( ′ ) ◦ . 

Then ψ ′ identifies ( ′ ) ◦ with R r′ in such a way that A V ∩ ( ′ ) ◦ is identified with 

A (V ∩X ′ ) ◦.Wethenhavea ∈ A V if and only if ψ ′(a) ∈ A (V ∩X ′ ) ◦. 

LEMMA 5.4 

The map ψ is uniformly continuous. 

i=1


Proof 

The projective embedding φ induces a map (C ∗ ) r ↩→ (C ∗ ) d−1 , which is defined by 

monomials. This induces a linear map from R r → A (C ∗ ) r ⊂ Rd−1 . 

We also have a map from the moment polytope d−1 of (P d−1 ) to which is the 

projection induced by the inclusion of tori T ⊂ T d−1 . 

The composite 

R r −→ R d−1 

ψ d−1 

−−−→ d−1 −→ 

is ψ . Since the first and last maps are uniformly continuous, it suffices to show that 

ψ d−1 is uniformly continuous. 

This is fairly straightforward. Write µ d−1 = (µ 1 ,...,µ d−1 ).Forz ∈ (P d−1 ) ◦ , 

we may write z = (1,z 1 ,...,z d ) and µ j (z) =|z j | 2 / ( 1 + ∑ d−1 

i=1 |z i| 2) .If|log|z j |− 

log|z 

j ′ ||


COROLLARY 5.6 

We have A I = ⋂ f ∈I A f . In particular, 

A I = { a ∈ ∣ ∣ f {a} is not lopsided, ∀f ∈ I 

} 

. 

It is noted that Corollary 5.6 holds for all toric varieties with a moment map, not just 

the compact ones. However, the statement of uniform convergence in Corollary 5.5 

does not hold in general for noncompact toric varieties. For example, if one considers 

the toric variety C r , with the standard moment map µ(z) = (|z 1 | 2 ,...,|z r | 2 )/2, the 

convergence of the family LA ˜f n 

is almost never uniform. One can even see this in 

the simple example f (z) = (1 − z 1 ) ···(1 − z r ). The failure is that Lemma 5.4 does 

not hold; the map log|x| ↦→ |x| 2 /2 is not uniformly continuous, and so the uniform 

convergence does not carry over. 

It is most unfortunate that Proposition 5.3 does not easily carry over to the compact 

case. The use of elimination theory appears to be well suited only to the study of (C ∗ ) r 

with its particular standard symplectic form. 

6. Tropical varieties 

In this section, we show that Theorem 2 is the analytic counterpart to a theorem for 

tropical varieties. We have already seen examples of tropical hypersurfaces. Tropical 

varieties, in general, can be thought of as a generalisation of amoebas, where one 

replaces the norm |·|: C → R with a valuation in some nonarchimedean field. For 

this reason, tropical varieties are also known as nonarchimedean amoebas. 

Let K be an algebraically closed field with valuation v. For our purposes, a 

valuation on K is a map v : K → R trop , which satisfies the following conditions: 

• v(xy) = v(x) ⊙ v(y); 

• v(x + y) ≤ v(x) ⊕ v(y). 

This differs from the usual definition of a valuation in two purely cosmetic ways. First, 

a valuation is traditionally given as a map to v : K → R; we have simply translated it 

into the operations of R trop . Second, this is (−1) times the usual notion of a valuation. 

Our reasons for making these cosmetic changes becomes abundantly clear by the end 

of this section. 

To every f ∈ K[z 1 ,z −1 

1 ,...,z r,zr 

−1 ], we can associate a tropical polynomial as 

follows. If f = ∑ −→ b−→ k ∈A k 

z k 1 

1 ···zk r 

r , write 

f τ (x) = ⊕ −→ k ∈A 

v(b−→ k 

) ⊙ x −→ k 

= max 

−→ k ∈A 

{ 

v(b −→ k 

) + x · −→ k } ,


and call it the tropicalisation of f . We denote the tropical hypersurface associated to 

f τ by T f . 

If a ∈ R r trop 

, we can assign a weight to every monomial m ∈ 

K[z 1 ,z −1 

1 ,...,z r,z −1 ]. Define the weight of m at a to be 

r 

wt a (m) := m τ (a). 

If f (z) = ∑ d 

i=1 m i(z), where m i are monomials, let 

f {a} τ = { wt a (m 1 ),...,wt a (m d ) } . 

Recall that in R trop , a list of numbers {b 1 ,...,b r } is (tropically) lopsided if the 

maximum element of this list does not occur twice (in which case, the maximum 

element is greater than the tropical sum of all the other elements). Thus f {a} τ is 

lopsided if and only if a /∈ T f . 

Let I ⊂ C[z 1 ,z −1 

1 ,...,z r,zr 

−1 ] be an ideal, and let 

V = V (I) = { z ∈ (K ∗ ) ∣ r f (z) = 0, ∀f ∈ I } 

be its affine variety. Let val :(K ∗ ) r → R r trop 

be the map 

val(z) = ( v(z 1 ),...,v(z n ) ) . 

The following theorem, as stated, most closely resembles the formulation in [SS], 

though variants of it have also appeared in [EKL]and[St]. 

THEOREM 3 (Speyer and Sturmfels [SS, Theorem 2.1]) 

The following subsets of R r trop coincide: 

(1) the closure of the set val(V ); 

(2) the intersection of all tropical hypersurfaces ⋂ f ∈I T f j; and 

(3) the set of points a ∈ R r trop such that f {a} τ is not lopsided for all f ∈ I. 

This set is called the tropical variety of the ideal I. 

In fact, a stronger result than Theorem 3 (as stated here) is shown in [SS]. Let 

k denote the residue field of K. IfI ⊂ K[z 1 ,...,z r ], then one can construct an 

initial ideal of I, in k[z 1 ,...,z r ], corresponding to any weight a ∈ R trop . One can 

equivalently describe the tropical variety of I as the set of points a ∈ R trop such 

that the associated initial ideal contains no monomial. Thus the tropical variety is 

a subcomplex of the Gröbner complex, and there are algorithms to compute it (see 

[BJS+]).


One can easily see that Theorem 3 is precisely analogous to the summary given in 

Remark 5.1. The proofs of these results, however, are extremely different. An obvious 

question, therefore, is whether analogous statements can be made in other contexts. 

The following is a general context in which one may hope for such a theorem to 

be true. Suppose that K is an algebraically closed field, and let S(⊙, ⊕, ≤) be a totally 

ordered semiring. Suppose that ‖·‖ K : K ∗ → S satisfies the following conditions: 

(1) ‖xy‖ =‖x‖ K ⊙‖y‖ K for all x,y ∈ K; 

(2) for all a,b ∈ S, wehave 

a ⊕ b = max { ‖x + y‖ K 

∣ ∣ ‖x‖K = a, ‖y‖ K = b } . 

In particular, condition (2) implies that ‖x + y‖ K ≤‖x‖ K ⊕‖y‖ K for all x,y ∈ K. 

Thus ‖·‖ K is an S-valued norm. 

Let f ∈ K[z 1 ,z −1 

1 ,...,z r,zr 

−1 ], and write f = ∑ d 

i=1 m i as a sum of monomials. 

For any point a ∈ S r ,letζ be such that ‖ζ‖ K = a. Wedefine 

f {a} := { ‖m 1 (ζ)‖ K ,...,‖m d (ζ)‖ K 

} 

. 

As ‖·‖ K is multiplicative, this is independent of the choice of ζ. SinceS is totally 

ordered, we can define a list of elements of S to be lopsided if and only if one number 

is greater than the sum of all the others. 

Let V ⊂ (K ∗ ) r be a variety defined by an ideal I ⊂ K[z 1 ,z −1 

1 ,...,z r,zr 

−1 ].We 

can consider the following sets: 

• The closure of {(‖z 1 ‖ K ,...,‖z r ‖ K ) | z ∈ V }; 

• {a ∈ S r | f {a} is not lopsided, ∀f ∈ I}. 

The question is whether these two sets are equal for a particular (K,S,‖·‖ K ). 

In this article, we primarily discuss the example in which K = C, S = R + ,and 

‖·‖ C =|·|, and we show that they are equal. We have also just seen that this is true 

if K is a nonarchimedean field with ‖·‖ K as its valuation and S = R trop . 

Many (though not quite all) of the elements of the proof of Theorem 1 are valid in 

a more general context. Suppose that in addition to being a totally ordered semiring, 

S is a Q + -module (i.e., we can make sense of such things as (2/3)a for a ∈ S). For 

example, R trop is a Q + -module with trivial Q + -action. 

Define a binary operation ⊖ on S by 

a ⊖ b := min{c ∈ S | c ⊕ b ≥ a} 

whenever this set is nonempty. (We need not overly concern ourselves with the fact that 

a precise minimum may not exist: one can always get around the problem by treating 

this set as a Dedekind cut.) Then the triangle inequality ‖x − y‖ K ≥‖x‖ K ⊖‖y‖ K 

is valid (assuming that ‖x‖ K ≥‖y‖ K ). To see this, note that a ≤ a ′ implies that


a ⊖ b ≤ a ′ ⊖ b; thus 

‖x‖ K ⊖‖y‖ K ≤ (‖x − y‖ K ⊕‖y‖ K ) ⊖‖y‖ K . 

Clearly, ‖x − y‖ K ∈{c ∈ S | c ⊕‖y‖ K ≤‖x − y‖ K ⊕‖y‖ K }, which implies that 

‖x − y‖ K ≥ (‖x − y‖ K ⊕‖y‖ K ) ⊖‖y‖ K . 

A closer examination of the proofs of Lemma 2.1 and Calculation A.3 now reveal 

that they are also valid (almost word for word) for a general (K,S,‖·‖ K ). We can 

also prove Lemma 3.5, in general, by replacing the integral over the torus 

1 

(2πi) r ∫|z 1 |=1 

∫ ( ∑ 

··· 

|z r |=1 

m−→ j 

(e a 1 z 1,...,e a r z r) 

−→ z l 1 

1 ···z l r 

r 

j 

by a discrete average over a finite subgroup of the torus 

1 

N r 

) dz1 

z 1 ···dz 1 

z 1 

∑ ∑ ( ∑ m−→ j 

(e a 1 

··· 

z 1,...,e a r z r) ) 

. 

z 1 :z1 N =1 z r :zr N =1 −→ z l 1 

1 ···z l r 

r 

j 

If N is suitably large, this discrete average has the same effect as the integral (i.e., 

picking out a single term from the polynomial). In fact, we can follow the proof of 

Theorem 1(1), up to and including inequality (3.5). All that remains is to show that 

the right-hand side of (3.5) becomes sufficiently small as n gets large. Unfortunately, 

in general, this is not always true for all γ


(1) c 1 n D 1 ((1 + γ n ) c 0n D 0 

− 1) < 1/2; 

(2) (1 + γ n ) c 0n D 0 

< 3/2. 

Proof 

We have 

Also, 

( (8 ) 

n log γ −1 ≥ D 0 + D 1 log n + log 0 c 1 

3)c 

⇔ γ −n ≥ 

( 8 

3) 

c 0 n D 0 

c 1 n D 1 

⇔ c 0 n D 0 

γ n 3 

≤ . (A.1) 

8c 1 n D 1 

3 

( 

1 

) 

≤ − 1 ( 

1 

) 2 

8c 1 n D 1 2c 1 n D 1 2 2c 1 n D 1 

( 

< log 1 + 1 ) 

2c 1 n D 1 

(A.2) 

( 3 

< log . 

2) 

(A.3) 

Using (A.1), (A.2), and the fact that log(1 + γ n )


CALCULATION A.2 

Assume that c ≥ 1, 0


Putting together (A.6)and(A.8), 

e (r+2)cnr−1 γ n − 1 

< (2−1−r )(2+ r)/(2 + r + 2 −1−r ) 

2 − e cnr−1 γ n (2 + r)/(2 + r + 2 −1−r ) 

= 2 −1−r . 

CALCULATION A.3 

For x>0 and s ∈ Z + , 

∑ 

( ) 

w0 + s − 1 ∑ x w 

s − 1 w!


References 

[A] M. AUDIN, The Topology of Torus Actions on Symplectic Manifolds, Progr. Math. 93, 

Birkhäuser, Basel, 1991. MR 1106194 433, 435 

[BM] U. BETKE and P. MCMULLEN, Lattice points in lattice polytopes, Monatsh. Math. 99 

(1985), 253 – 265. MR 0799674 418 

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tropical varieties, J. Symbolic Comput. 42 (2007), 54 – 73. MR 2284285 439 

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185, Springer, New York, 1998. MR 1639811 434 

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varieties, preprint, arXiv:math/0408311v2 [math.AG] 411, 439 

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hyperplane amoebas, Adv. Math. 151 (2000), 45 – 70. MR 1752241 426, 428 

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Press, Princeton, 1993. MR 1234037 435 

[GKZ] I. M. GEL’FAND, M. M. KAPRANOV, andA. V. ZELEVINSKY, Discriminants, 

Resultants, and Multidimensional Determinants, Math. Theory Appl., Birkhäuser, 

Boston, 1994. MR 1264417 407, 436 

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Faces of Geometry, Int. Math. Ser. (N.Y.) 3, Kluwer/Plenum, New York, 2004, 

257 – 300. MR 2102998 408 

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triangulations of the Newton polytope, Duke Math. J. 121 (2004), 481 – 507. 

MR 2040284 429 

[R] L. I. RONKIN, “On zeros of almost periodic functions generated by functions 

holomorphic in a multicircular domain” (in Russian) in Complex Analysis in 

Modern Mathematics, FAZIS, Moscow, 2001, 239 – 251. MR 1833516 429 

[Ru] H. RULLGÅRD, Polynomial amoebas and convexity, preprint, 2001. 429 

[S] D. SPEYER, personal communication, 2003. 410 

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389 – 411. MR 2071813 411, 439 

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MR 1969643 408, 428 

Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario 

N2L 3G1, Canada; kpurbhoo@math.uwaterloo.ca

ENDOSCOPIC LIFTING IN CLASSICAL GROUPS 

AND POLES OF TENSOR L-FUNCTIONS 

DAVID GINZBURG 

Abstract 

In this article, we introduce a new construction of endoscopic lifting in classical 

groups. To do that, we study a certain small representation and use it as a kernel 

function to construct the liftings. As an application of the construction, we study 

the relations of poles of tensor L-function with certain liftings and certain period 

integrals. 

Contents 

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 

2. Notation and basic definitions . . . . . . . . . . . . . . . . . . . . . . 450 

3. The cuspidality of the lift . . . . . . . . . . . . . . . . . . . . . . . . . 465 

4. The nonvanishing of the lift . . . . . . . . . . . . . . . . . . . . . . . . 474 

5. The unramified computations . . . . . . . . . . . . . . . . . . . . . . . 484 

6. Liftings and poles of tensor L-functions . . . . . . . . . . . . . . . . . 488 

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 


This article presents a new construction for endoscopic liftings for classical groups. We 

consider five cases, all of which are described in Definition 2. For an example of this 

construction, consider the following case. Corresponding to the homomorphism of L- 

groups SO 2n+1 (C)×SO 2m (C) ↦→ SO 2(n+m)+1 (C), the Langlands conjectures predict a 

lifting from automorphic representations of the group Sp 2n (A)×SO 2m (A) to automorphic 

representations of Sp 2(n+m) (A). Thus, our goal in this article is the following: given 

two cuspidal generic irreducible representations on the groups Sp 2n (A) and SO 2m (A), 

we construct a generic cuspidal representation defined on Sp 2(n+m) (A) which corresponds 

to the above lifting. These types of liftings are examples of what is known as 

endoscopic lifting. Other examples of these liftings were constructed in [GRS7] using 

the descent method (for more on L-functions and liftings, see [L1], [A], [B]). 


Vol. 141, No. 3, c○ 2008 DOI 10.1215/00127094-2007-002 

Received 12 October 2006. Revision received 18 May 2007. 

2000 Mathematics Subject Classification. Primary 11F70; Secondary 22E55. 

Author’s work partially supported by Israel Science Foundation grant 162/06. 

447

448 DAVID GINZBURG 

The method we use to construct these liftings is what we referred to in [G2]asthe 

theta lifting method. By that we mean that we construct a representation, specifically 

a residue of a certain Eisenstein series, defined on a certain group M(A) and then use 

it as a kernel function in order to construct our lifting. For example, in the above case, 

we start with a cuspidal representation τ = τ(ɛ) of the group GL 2m (A) whichisa 

functorial lift from a cuspidal representation ɛ of SO 2m (A) as constructed in [CKPS]. 

We then use this representation to construct a residue representation ɛ defined on the 

group M(A) = Sp 2m(2n+1) (A). Using a suitable unipotent integration, we obtain a copy 

of the group Sp 2n × Sp 2(n+m) embedded inside M. Starting with a generic cuspidal 

representation σ of Sp 2n (A), we pair it against the above kernel representation, thus 

obtaining an automorphic representation defined on Sp 2(n+m) (A). More precisely, the 

automorphic representation that we obtain is the space generated by all functions of 

the form 

∫ ∫ 

( ) 

f (h) = 

ϕ σ (g)θ ɛ u(g, h) ψU (u) dudg. 

Sp 2n (F )\Sp 2n (A) U(F )\U(A) 

Here U is a unipotent group with a character ψ U defined on it. The functions ϕ σ and θ ɛ 

are vectors in the representation space of σ and ɛ , respectively; also, h ∈ Sp 2(n+m) (A) 

(for more details, see Section 2.3). It should be mentioned that our construction works 

if we replace the cuspidal representation ɛ on GL 2m (A) with a generic automorphic 

representation that is a constituent of an induced representation from cuspidal data. 

Sections 3 – 5 contain the setup of the construction and the basic properties of the 

lifting. Since the ideas behind the proofs are similar in all cases, we concentrate on one 

example, namely, the example mentioned above. Since most of the proofs of the basic 

properties are now quite standard, we have allowed ourselves a certain sketchiness in 

some of the technical details. 

Section 2.2 is devoted to the construction of the representation ɛ and the study 

of its basic properties. As mentioned above, this representation is defined as a residue 

of an Eisenstein series. In Section 2.3, we define the various lifts that we intend to 

construct and set up the global integral that we use for this construction. Section 3 is 

devoted to the proof of the cuspidality of the lift; this is done via direct calculations 

of the various constant terms. As usual in these types of construction, there is an 

obstruction for the lift to be cuspidal (see the beginning of Section 3 for more details). 

In Section 4, we prove the nonvanishing of the lift by showing that the representation 

of Sp 2(n+m) (A) obtained by the above integral is, in fact, a generic representation, 

under the assumption that σ is generic. As a consequence of Theorem 3, the main 

theorem in that section, we obtain that the lift is indeed nonzero. Section 5 is devoted 

to proving that the lift constructed is indeed functorial. We do this by computing the 

standard local L-function of the lift using the basic identity we derived in Theorem 3.

ENDOSCOPIC LIFTING 449 

In the last section, Section 6, we apply our construction to relate liftings with 

period integrals and poles of L-functions. We consider two cases. In the first case, we 

characterize the existence of a simple pole to standard tensor L-functions. (The main 

statement is Conjecture 1, together with Theorem 6.) More precisely, we prove the 

following theorem. 

MAIN THEOREM 

Let π denote an irreducible generic cuspidal representation of the orthogonal group 

SO 2(n+m)+1 (A), and let ɛ denote an irreducible generic cuspidal representation of 

SO 2m+1 (A). Letτ(ɛ) denote the lift of ɛ to GL 2m (A) whose existence was proved 

in [CKPS]. Assume that τ(ɛ) is a cuspidal representation. Then the following are 

equivalent. 

(1) The partial tensor L-function L S (π × ɛ, s) = L S (π × τ(ɛ),s) has a simple 

pole at s = 1. HereS is a finite set of places including the archimedean ones 

such that outside of S, all data is unramified. 

(2) There is a choice of data such that the period integral P(π, τ(ɛ)), defined in 

the proof of Theorem 6, is not zero for some choice of data. 

(3) There is a generic cuspidal representation σ of SO 2n+1 (A) such that π is the 

weak endoscopic lift from σ and ɛ. 

Two implications of this theorem follow from other references (see the first paragraph 

following Conjecture 1). The implication that statement (2) implies statement (3) is 

proved in Theorem 6 using our construction of the lifting. 

The second application that we discuss in Section 6 is more conjectural. Motivated 

by the low-rank cases that we describe in that section, we give a conjecture as to when 

the L-function associated to the tensor product of two Spin representations can have 

poles at certain values. Although the main conjecture is stated in Conjecture 3, we 

have more evidence in a special case of that conjecture stated in this form. 

CONJECTURE 2 

Let π and ɛ denote irreducible cuspidal generic representations of the groups 

GSO 2(m+4) (A) and GSp 2m (A). The following statements are equivalent. 

(1) The partial L-function L S (π × ɛ, Spin 2(m+4) × Spin 2m+1 ,s) has a simple pole 

at s = 1. 

(2) The period integral Q(π, ɛ), described in Section 6, is not zero for some choice 

of data. 

(3) There exists a cuspidal generic representation ν of the exceptional group G 2 (A) 

such that π is the weak functorial lift from ν and from ɛ. 

In other words, this conjecture ties the poles of the L-function of a tensor product 

of Spin representations, with a certain period integral and with a lifting related to 

the exceptional group G 2 . Indeed, Conjecture 2 is related to the following lifting.


Let ν denote a cuspidal automorphic representation of the exceptional group G 2 (A), 

and let ɛ denote a cuspidal representation of GSp 2m (A). Corresponding to the L- 

groups homomorphism G 2 (C) × GSpin 2m+1 (C) ↦→ GSpin 2m+8 (C), the Langlands 

conjectures predict a lifting from ν and ɛ to an automorphic representation defined 

on the group GSO 2m+8 (A). This lifting is not an endoscopic lifting. As we state in 

Theorem 9, Conjecture 2 is, in fact, a theorem in the case where m = 1; thiswas 

provedin[GH2]. Our contribution to this conjecture is the implication that statement 

(2) implies (3), which we prove in Theorem 10. To prove this theorem, we use our 

construction of the lifting and extend it to similitude groups. 

It is worth mentioning that the conjectures that we make and the theorems that 

we prove are also important in the study of the Langlands conjectures. These results 

tie the existence of poles of L-functions to a certain lifting. In addition, the image 

of the lift is characterized by a certain period integral. To prove these results, it is 

necessary to combine the so-called Rankin-Selberg method together with what we 

designate the theta lifting method. In our study of Conjecture 1, the fact that the 

corresponding L-function had an integral representation using the Rankin-Selberg 

method is crucial; Eulerian Rankin-Selberg integrals are a relatively rare phenomena. 

Indeed, the L-functions studied in Conjecture 2 do not have, as far as we know, 

such an integral representation except when m = 1. All this indicates that in order 

to study such conjectures, one also has to consider Rankin-Selberg integrals that 

are not Eulerian. It is conceivable that the study of residues of L-functions can be 

done by non-Eulerian integrals. This is clearly a step toward studying conjectures of 

the type stated in Conjecture 2. Another example of this phenomena was studied in 

[BFG, page 290], where a global period is given which conjecturally characterizes 

the image of the cubic Shimura lift for the group PGL 3 . The given period integral 

involves the minimal representation of the group SO 8 , which is a residue of a degenerate 

Eisenstein series. A possible way to study this period integral is to replace this 

minimal representation with the degenerate Eisenstein series and to unfold the integral. 

This way, one obtains a non-Eulerian integral whose residue, at least conjecturally, 

characterizes a possible lift. 

Finally, we point out two possible extensions of our construction. First, we can 

consider representations σ that are not generic. Another possible extension is to replace 

the representation ɛ with a representation that is nearly equivalent to it. In both cases, 

we expect at least some of the results stated in this article to still be valid. We hope to 

address these cases in the near future. 

2. Notation and basic definitions 

2.1. General notation 

Let M denote a split classical group of type B,C, orD. In terms of matrices, 

we represent these groups with respect to the following forms. Let J n denote the


(n × n)-matrix, with 1’s along the second diagonal. For the orthogonal groups, we 

use the form corresponding to the matrix J n for a suitable value of n. The symplectic 

groups are represented with respect to the form ( J n 

) 

−J n . For the symplectic groups, 

we denote Mat 0 r×r ={A ∈ Mat r×r : A t J n = J n A}. For orthogonal groups, we denote 

Mat 0 r×r ={A ∈ Mat r×r : A t J n =−J n A}. Given a matrix X ∈ Mat n×n , we denote 

X ∗ = J n (X t ) −1 X. 

When M = Sp 2n , we use ˜M to denote its double cover. 

Given a parabolic subgroup of M, we refer to it as a standard parabolic subgroup of 

M if it contains the Borel subgroup of M which consists of upper triangular matrices. 

Let F denote a global field, and let A denote its ring of adeles. Let denote an 

automorphic representation defined on M(A). 

Given a subgroup U of M, we denote 

∫ 

θ U (g) = θ(ug) du. 

U(F )\U(A) 

Here θ is a vector in the space of the representation . 

Let ψ denote a nontrivial additive character of the group F \A. Letψ U denote a 

character of the group U(F )\U(A). We denote 

∫ 

θ U,ψ U 

(g) = θ(ug)ψ U (u) du. 

U(F )\U(A) 

2.2. Definitions and properties of some small representations 

In this section, we define and study the properties of certain small representations in 

the symplectic and split orthogonal groups obtained by residues of Eisenstein series. 

For basic definitions and properties of Eisenstein series, we refer the reader to [L2]or 

[MW]. The results of this section are quite straightforward and follow [G2], [GRS9], 

and [GRS7] and the references listed in those articles. There are five cases to consider. 

We construct the representation that we need in one case, while the other cases are 

mentioned at the end of this section, and their proofs are similar. 

Let τ denote a cuspidal representation of GL 2m (A). In the notation of 

[G2, Section 2], our goal is to study the multivariable Eisenstein series E τ (g, ¯s) 

defined on the group Sp 4mn (A). The main result is the following. 

PROPOSITION 1 

The Eisenstein series has a simple pole at the point ¯s = ¯s 0 in the sense that the limit 

lim¯s→¯s0 (¯s − ¯s 0 )E τ (g, ¯s) is nonzero. Here ¯s 0 is defined as in [G2, Section 2]. As a 

function of g ∈ Sp 4mn (A), we denote this representation by τ .


Proof 

The proof comes by analyzing the constant term along the unipotent group V of 

the Eisenstein series. Recall from [G2] thatE τ (g, ¯s) is associated to the induced 

(τ ⊗···⊗τ)δ¯s¯Q . Here Q = (GL 2m ×···×GL 2m )V , where 

V is the unipotent radical of Q. Thus, in the notation of Section 2.1, we consider the 

constant term Eτ V (g, ¯s). The first statement is that 

representation Ind Sp 4mn (A) 

Q(A) 

E V τ (g, ¯s) = ∑ w∈W 0 

M w (f τ , ¯s)(g), 

where W 0 and M w are defined as in [G2, (2.3)] and in the text following that formula. 

To prove this, we argue as in [GRS7, Sections 2.1, 2.2]. In fact, our Eisenstein series 

is a special case of the Eisenstein series considered in [GRS7, Section 2]. The point 

is that for any Weyl element w not in W 0 , when considering the conjugation wV w −1 , 

we end up integrating the cuspidal representation τ along a unipotent radical of a 

parabolic subgroup of GL 2m . Hence we get zero. 

In terms of matrices, the set W 0 can be described as follows. Let ˜W denote the 

Weyl group of Sp 2n ; in matrices, we may choose a set of representatives to be all 

permutation matrices of Sp 2n whose nonzero elements consist of 1’s and –1’s. Take 

such an element, and replace each 1 by the identity matrix of size 2m; follow a similar 

process with each −1. 

Arguing as in [G2, (2.4) – (2.11)], we deduce that the following limits lim¯s→¯s0 (¯s − 

¯s 0 )M w (f τ , ¯s)(g) are, in fact, zero, except when we take w, which corresponds to the 

long Weyl element in ˜W. In other words, the above limits are all zero, except when 

we take w to have blocks of identity matrix of size 2m on the second diagonal. In this 

case, we obtain from [G2, (2.9)] the fact that M w (f τ , ¯s)(g) is equal to 

∏ 

n∏ L S (τ,ρ i ,ζ i ) 

M w,ν (f τν , ¯s)(g ν ) 

L S (τ,ρ i ,ζ i + 1) LS τ (¯s), 

ν∈S 

i=1 

where the notation is as defined in [G2, (2.10), (2.11)]. Computing the limit in this 

case, we get a nonzero factor. 

 

The next step is to determine which Fourier coefficients the representation τ supports 

and which it does not. This is best expressed in terms of the structure of the unipotent 

orbits associated with the group Sp 4mn . The idea of relating representations with 

unipotent orbits is not new (e.g., see [Sp]). We mainly use the description of unipotent 

orbits as given in [CM]. The way in which we associate unipotent orbits with Fourier 

coefficients is explained in detail in [G3]; we do not review it here. As in [G2, 

Definition 3], we have the following.


Definition 1 

Let π denote an automorphic representation of G = Sp 4mn . We denote by O G (π) 

the set of all unipotent classes of H with the following property. A unipotent class 

O ∈ O G (π) if, for all unipotent classes Õ > O, the representation π does not have a 

nonzero Fourier coefficient corresponding to Õ. When there is no confusion, we write 

O(π) for O G (π). 

The main result of Section 2 is the following theorem. 

THEOREM 1 

We have O( τ ) = ((2m) 2n ). 

Proof 

To prove this theorem, we need to prove two things. First, we need to prove that given a 

unipotent orbit O of Sp 4mn which is greater or not related to ((2m) 2n ), the representation 

τ has no nonzero Fourier coefficient that corresponds to O. Then we need to show 

that τ has a nonzero Fourier coefficient that corresponds to the unipotent orbit 

((2m) 2n ). The first part follows as in [GRS9, Sections 2, 3], but by replacing n with 

m and k with n. Indeed, the proof is local. We show that the unramified constituents 

of the residue cannot support any functional that is induced from a global Fourier 

coefficient corresponding to a unipotent orbit that is bigger than or not related to 

((2m) 2n ). It follows from [GRS9, Definition 2.1, Lemma 3.1] that the local unramified 

constituents are the same as in our case defined above. By applying [GRS9, Lemma 3.3, 

Proposition 3.6], we may deduce that a similar result holds for τ defined here. 

Thus we need only prove the second part, namely, that τ has a nonzero Fourier 

coefficient that corresponds to the unipotent orbit ((2m) 2n ).Let̂Q denote the standard 

parabolic subgroup of Sp 4mn whose Levi part is GL 2n ×···×GL 2n .LetU denote its 

unipotent radical. In terms of matrices, we have 

⎧⎛ 

⎞ 

⎫ 

I 2n X 1 ∗ 

I 2n X 2 ∗ 

. .. 

⎪⎨ 

Xm−1 ∗ 

U = 

I 2n Y 

⎪⎬ 

I 2n Xm−1 

∗ : X i ∈ Mat 2n×2n ,Y∈ Mat 0 2n×2n . 

. 

⎜ 

.. ⎟ 

⎝ 

⎪⎩ 

I 2n X1 

∗ ⎠ 

⎪⎭ 

I 2n 

On U(F )\U(A), we define a character ψ U as follows. For Y = (Y i,j ),defineψ U (u) = 

ψ(tr(X 1 +···+X m−1 ) + tr ′ (Y )), where tr ′ (Y ) = Y 1,1 +···+Y n,n . 

For a vector θ τ in the space of τ , consider the Fourier coefficient θ U,ψ U 

τ 

(g). 

Following [G3], it is not hard to check that this Fourier coefficient does indeed


correspond to the unipotent orbit ((2m) 2n ). One can also check that the stabilizer 

inside the Levi part of ̂Q is the group SO 2m , which is the stabilizer of this unipotent 

orbit (see [CM, Theorem 6.1.3], which is due to Springer and Steinberg). Our goal is 

to prove that θ U,ψ U 

τ 

(g) is nonzero for some choice of data. We assume that it is zero 

for every choice of data, and we derive a contradiction. 

Let w denote the Weyl element defined as follows. For 1 ≤ i ≤ 2m and 0 ≤ j ≤ 

2n − 1,the(2mj + i, 2n(i − 1) + j)-entry of w is ±1, and all other entries are zero. 

The entry is −1 if 2mj + i>2mn and 2n(i − 1) + jiand z i,i = I 2m . Finally, for i>j, the matrix z i,j is upper unipotent 

with zero along the diagonal. Next, the matrix r 4mn = (r i,j ), where r i,j ∈ Mat 2m×2m , 

is such that r i,j = 0 if i>j,andr i,i = I 2m .Fori


and only if the integral 

∫ 

θ τ (vy 4mn )ψ m (y 4mn ) dv dy 4mn 

is zero for every choice of data. 

Recall that V is the unipotent radical of the parabolic subgroup Q which was 

defined in the beginning of the proof of Proposition 1. Arguing as in [GRS5, Theorem 1, 

pages 889 – 894], we conclude that the above Fourier coefficient is nonzero for some 

choice of data. Thus, we obtain a contradiction. 

 

We also need the following local result. 

PROPOSITION 2 

Let F be a nonarchimedean local field. For this proposition only, let θ τ denote the 

local constituent of τ at the place F . Assume that θ τ is unramified. Let l denote a 

functional defined on θ τ which satisfies the property l(vy 4mn ω) = ψm −1(y 

4mn)l(ω) for 

all v ∈ V , for all y 4mn that were defined in the proof of Theorem 1, and for all vectors 

ω ∈ θ τ . Then the space of such functionals is 1-dimensional. 

Proof 

Let L denote the maximal unipotent subgroup of Sp 4mn . Thus, every element in L 

has a unique factorization as vy 4mn .From[G2, Lemma 3.1], we deduce that θ τ is a 

quotient of the induced representation Ind Sp 4mn 

̂Q (χ 1 ⊗···⊗χ m ) δ 1/2 . Here ̂Q 1 

̂Q is the 

standard parabolic subgroup of Sp 4mn whose Levi part is GL 2n ×···×GL 2n ,and 

χ i are certain unramified characters. To prove the proposition, we apply the Bruhat 

theory; thus, it is enough to prove the following. We say that g ∈ ̂Q\Sp 4mn /L is 

not admissible if we can find an element y 4mn as above, so that ψ m (y 4mn ) ≠ 1 and 

gy 4mn g −1 ∈ ̂Q. Otherwise, we say that g is admissible. To prove the proposition, it is 

enough to show that there is at most one admissible double coset. This proves that the 

space of such functionals is at most 1-dimensional. From Theorem 1, it follows that 

the space of such functionals is exactly 1-dimensional. 

We can choose elements in ̂Q\Sp 4mn /L as Weyl elements modulo from the left by 

Weyl elements in GL 2n ×···×GL 2n . We choose the Weyl elements to be permutation 

matrices. Let w be such a Weyl element. Consider the first n rows of w. We claim that 

for w to be admissible, we must have w i,2ni ≠ 0 whenever 1 ≤ i ≤ n. Indeed, if not, 

then from the definition of ψ m , we can find a matrix in y 4mn so that ψ m (y 4mn ) ≠ 0 

and wy 4mn w −1 ∈ ̂Q. Indeed,iffor1 ≤ i ≤ n we have w i,j ≠ 0 and j ≠ 2ni, then 

the matrix x(r) = I + re j,j+1 satisfies wx(r)w −1 ∈ ̂Q and ψ(x(r)) ≠ 1. Here I is 

the (4mn)-identity matrix, and e p,q is the (4mn)-matrix with 1 at the (p, q)-entry and 

zero elsewhere.


Next, consider the rows 2n + i for 1 ≤ i ≤ 2n. We claim that if w is admissible, 

then w 2n+i,2ni−1 ≠ 0. Indeed, suppose that w 2n+i,j ≠ 0 for some 1 ≤ i ≤ 2n and 

that j ≠ 2ni − 1. Then, using the Weyl group of GL 2n ×···×GL 2n if needed, we 

can find l>2n + i so that w l,j+1 ≠ 0. This means that x(r) = I + re j,j+1 satisfies 

wx(r)w −1 ∈ ̂Q.Sincex(r) is in L, and since ψ m (x(r)) ≠ 1,thenw is not admissible. 

Thus w 2n+i,2ni−1 ≠ 0 for all 1 ≤ i ≤ 2n. Continuing by induction, we see that if w is 

admissible, then w is also uniquely determined. 

 

Returning to the global situation, the following proposition follows immediately from 

Theorem 1. 

PROPOSITION 3 

Let θ U,ψ U 

τ 

(g) denote the Fourier coefficient corresponding to the unipotent orbit 

((2m) 2n ), as was described in Theorem 1. Letg ∈ SO 2n (A), which are the adelic 

points of the stabilizer of this Fourier coefficient (see the proof of Theorem 1). Then 

θ U,ψ U 

τ 

(g) = θ U,ψ U 

τ 

(e). In other words, θ U,ψ U 

τ 

(g) is invariant under all g ∈ SO 2n (A). 

Proof 

The idea is similar to the one sketched in [GRS8, Theorem 2.1]. Assume first that 

n ≥ 2. As mentioned above, the stabilizer of the character ψ U is the split orthogonal 

group SO 2n (A).Letx(r) denote the 1-parameter subgroup of SO 2n (A) corresponding 

to the highest-weight root vector in this group. If θ U,ψ U 

τ 

(g) were not left-invariant 

under elements g ∈ SO 2n (A), it would follow that for some a ∈ F ∗ , the integral 

∫ 

F \A 

( ) 

θ U,ψ U 

τ x(r)g ψ(ar) dr 

is not zero for some choice of data. But as mentioned in [GRS8], this last integral 

corresponds to a unipotent orbit that is strictly greater than ((2m) 2n ). Indeed, to show 

this, let U 0 denote the unipotent subgroup of U defined as follows. Let u = (u i,j ) ∈ U. 

We consider all matrices u ∈ U so that u i,j = 0 for all pairs (i, j) ∈{(2nk − 1, 2nk + 

1), (2nk − 1, 2nk + 2), (2nk, 2nk + 1), (2nk, 2nk + 2) : 1 ≤ k ≤ m}. If we restrict 

ψ U to U 0 , we obtain a character of U 0 (F )\U 0 (A) which we continue to denote by ψ U . 

Next, we define another unipotent subgroup of Sp 4mn , which we denote by U 1 , 

which contains U 0 . We define this group by the unipotent group generated by U 0 , 

and all 1-parameter unipotent matrices I 4mn + re 

i,j ′ , where (i, j) ∈{(2nk + 1, 2nl − 

2), (2nk + 1, 2nl − 1), (2nk + 2, 2nl − 2), (2nk + 2, 2nl − 1) : 0 ≤ k ≤ m − 1; 1 ≤ 

l ≤ m},ande 

i,j ′ = e i,j − e 4mk−j+1,4mk−i+1 . 

Consider the above integral. We now perform certain Fourier expansions. Let 

y(r 1 ,r 2 ,r 3 ,r 4 ) = I 4mn + r 1 e ′ 1,2n−1 + r 2e ′ 1,2n + r 3e ′ 2,2n−1 + r 4e ′ 2,2n ,


and let 

z(r 1 ,r 2 ,r 3 ,r 4 ) = I 4mn + r 1 e ′ 2n−1,2n+1 + r 2e ′ 2n,2n+1 + r 3e ′ 2n−1,2n+2 + r 4e ′ 2n,2n+2 . 

Notice that the group of matrices generated by z(r 1 ,r 2 ,r 3 ,r 4 ) is a subgroup of U. 

Expanding the above integral along the group y(r 1 ,r 2 ,r 3 ,r 4 ) with r i integrated over 

points in A modulo points in F , we obtain 

∑ 

∫ 

θ U,ψ U 

τ 

α i ∈F 

(F \A) 5 

( 

x(r)y(r1 ,r 2 ,r 3 ,r 4 )g ) ψ( 

ar + 

4∑ ) 

α i r i dr dr i . 

From the fact that θ τ (g) is an automorphic function, it follows that it is left-invariant 

under the rational points. Using that, conjugating with the matrices z(α 1 ,α 2 ,α 3 ,α 4 ) 

from left to right, and collapsing summation with integration, the above integral is 

equal to 

∫ 

∫ 

A 4 F \A 

i=1 

θ U 1,1,ψ U 

( 

τ x(r)z(r1 ,r 2 ,r 3 ,r 4 )g ) ψ(ar) dr dr i . 

Here U 1,1 is the unipotent subgroup of U 1 generated by all unipotent matrices of the 

form I 4mn + re 

i,j ′ , where (i, j) ∈{(2n − 1, 2n + 1), (2n − 1, 2n + 2), (2n, 2n + 

1), (2n, 2n + 2)} and the subgroup of U which consists of all matrices u = (u i,j ) ∈ U 

such that u i,j = 0 for all (i, j) ∈{(1, 2n − 1), (1, 2n), (2, 2n − 1), (2, 2n)}. Clearly, 

it is enough to prove the vanishing of the inner integral in the above integration. We 

continue this process inductively, along the corresponding subgroups of U 0 and U 1 , 

and we finally obtain that it is enough to prove the vanishing of the integral 

∫ 

θ U 1,ψ U 

( ) 

τ x(r)g ψ(ar) dr, 

F \A 

where we view ψ U as a character of U 1 by its restriction to U 0 . It follows from the 

definition of the correspondence between unipotent orbits and Fourier coefficients, as 

explained in [GRS8] or[G3], that this last integral is associated with the unipotent 

orbit ((2m + 1) 2 (2m) 2n−4 (2m − 1) 2 ). This unipotent orbit is, of course, greater than 

((2m) 2n ). Hence the above integral is zero for every choice of data. This produces a 

contradiction; therefore, the above left-invariant property holds. 

The case where n = 1 is still true but treated differently. Indeed, in that case, the 

stabilizer of ψ U is just a torus, and by using certain Fourier expansions, similar to the 

ones performed above, we obtain our result.


To state the following lemma, let N m denote the standard unipotent radical subgroup 

of the maximal parabolic of Sp 4mn whose Levi part is GL 2m ×Sp 4m(n−1) .LetV (GL 2m ) 

denote the standard maximal unipotent subgroup of GL 2m .Wedenotebyψ V (GL2m ) the 

Whittaker character of V (GL 2m ).LetN 0 m denote the subgroup of N m defined as 

⎧⎛ 

⎞ 

⎨ I 2m Y Z 

N 0 m = ⎝ I 

⎩ 

4m(n−1) Y ∗ ⎠,Y ∈ Mat 2m×4m(n−1) : Y 2m,i = 0, 

I 2m 

⎫ 

⎬ 

Z ∈ Mat 0 2m×2m : Z 2m,1 = 0 

⎭ . 

Denote 

⎛ 

X 

˜X = ⎝ 

I 4m(n−1) 

X ∗ ⎞ 

⎠, X ∈ V (GL 2m ). 

We have the following. 

LEMMA 1 

For every choice of data, we have the identity 

∫ ∫ 

θ τ (y˜X)ψ V (GL2m )(X) dy dX = 

Nm 0 (F )\N m 0 (A) ∫ ∫ 

N m (F )\N m (A) 

θ τ (y˜X)ψ V (GL2m )(X) dy dX, 

where X is integrated over V (GL 2m )(F )\V (GL 2m )(A). 

Proof 

We start by considering the Fourier expansion of the left-hand-side integral, along 

the unipotent group I 4mn + re 2m,4m(n−1)+1 . The contribution to the expansion from the 

nontrivial characters is zero. Indeed, each term corresponding to a nontrivial character 

corresponds to the unipotent orbit ((2m + 2)1 4mn−2m−2 ). From Theorem 1, it follows 

that this Fourier coefficient is zero. Thus we are left with only the trivial contribution. 

Hence, the left-hand side of the above identity equals 

∫ ∫ 

θ τ (y˜X)ψ V (GL2m )(X) dy dX, 

Nm 1 (F )\N m 1 (A) 

where now


⎧⎛ 

⎞ 

⎫ 

⎨ I 2m Y Z 

⎬ 

N 1 m = ⎝ I 

⎩ 

4m(n−1) Y ∗ ⎠,Y ∈ Mat 2m×4m(n−1) : Y 2m,i = 0,Z ∈ Mat 0 2m×2m⎭ . 

I 2m 

Next, we expand the above integral along the unipotent group N m /Nm 1 with points in A 

modulo points in F . This is an abelian group, and Sp 4m(n−1) (F ) acts on this expansion 

with two orbits. Here Sp 4m(n−1) is embedded in Sp 4mn as g ↦→ diag(I 2m ,g,I 2m ).The 

trivial orbit produces the integral on the right-hand side of the identity at the statement 

of the lemma. Thus we need to prove that the nontrivial orbit contributes zero. In other 

words, we need to prove that the integral 

∫ ∫ 

θ τ (y˜X)ψ V (GL2m )(X)ψ m (y) dy dX (1) 

N m (F )\N m (A) 

is zero for every choice of data. Here 

⎛⎛ 

⎞⎞ 

I 2m Y Z 

ψ m (y) = ψ m 

⎝⎝ 

I 4m(n−1) Y ∗ ⎠⎠ = ψ(Y 2m,1 ). 

I 2m 

This is done in a way similar to [GRS9, Lemma 3.3]. Indeed, if not zero, the above 

integral induces a local functional that is nonzero on each of the constituents of τ . 

However, as in the above reference, one can show that the local unramified constituent 

of τ cannot support such a functional. We omit the details. 

 

For the construction of the lifting defined in Section 2.3, we need to consider residues 

of other Eisenstein series which are defined on other classical groups. As we did 

above, we need to study their properties. Since the arguments are exactly the same, 

we define only the residues and indicate the corresponding unipotent orbit attached to 

these representations. We start with the following step. 

(1) Let G = Sp 2m(2n+1) .Letɛ denote a cuspidal generic irreducible representation 

of the split orthogonal group SO 2m (A). Letτ = τ(ɛ) denote the functorial lift of ɛ to 

GL 2m (A), aswasprovedin[CKPS]. We assume that τ is cuspidal. Let µ(ɛ) denote 

the lift of ɛ to Sp 2m given by the theta representation. It follows from [GRS2] that 

µ(ɛ) is generic. Let Q denote the standard parabolic subgroup of G whose Levi part 

is GL 2m ×···×GL 2m × Sp 2m . Here GL 2m occurs n times. Let E τ(ɛ),µ(ɛ) (g, ¯s) denote 

the Eisenstein series defined on G and associated with the induced representation 

Ind G(A) 

Q(A) (τ(ɛ) ⊗ ··· ⊗ τ(ɛ) ⊗ µ(ɛ))δ¯s Q . Write an element in the Levi part of Q as 

g = diag(g 1 ,...,g n ,h,gn ∗,...,g∗ 1 ). Then we define δ¯s Q = ∏ n 

i=1 |g i| s i δ1/2 Q (g). Asin 

the case of the Eisenstein series which we considered in Section 2.2, it is not hard to


check that up to a product of local intertwining operators, the poles of the Eisenstein 

series are determined by 

∏n−1 

i=1 

L S (τ × τ,s i − s i+1 ) L S (τ × µ(ɛ),s n ) 

L S (τ × τ,s i − s i+1 + 1) L S (τ × µ(ɛ),s n + 1) L τ (¯s), 

where L τ (¯s) is a product of partial L-functions that are holomorphic and do not vanish 

at the point s i = n − i + 1.Let¯s 0 = (n, n − 1,...,1). It follows from the above that 

the Eisenstein series has a pole at that point. 

If we denote by ɛ the residue of this Eisenstein series at the point ¯s 0 ,then 

O G ( ɛ ) = ((2m) 2n+1 ). Thus, the difference between this Eisenstein and the series that 

we studied at the beginning Section 2.2 is the use of the representation µ(ɛ) defined 

on Sp 2m (A). This difference is technical in its nature, and the statements proved above 

follow easily using similar arguments. 

(2) We have a similar situation on the double cover of the symplectic group. Denote 

G = ˜Sp 2m(2n+1) .Letɛ denote a generic cuspidal representation of ˜Sp 2m (A), andlet 

τ = τ(ɛ) denote the functorial lift to GL 2m from ɛ. By that we mean the following. It is 

well known (see, e.g., [GRS2]) that every generic cuspidal representation of ˜Sp 2m (A) 

has a functorial lift to a generic cuspidal representation of SO 2m+1 (A) or to a generic 

cuspidal representation of SO 2m−1 (A). Using the result of [CKPS], we can thus deduce 

that ɛ has a functorial lift to a cuspidal representation of GL 2m (A) or GL 2(m−1) (A). 

Henceforth, we assume that the given cuspidal representation ɛ has a functorial lift 

to a cuspidal representation τ = τ(ɛ) of GL 2m (A). In this context, we can therefore 

view the group Sp 2m (C) as the “L group” of ˜Sp 2m (see also [S]). 

We can form the Eisenstein series Ẽ τ(ɛ),ɛ (g, ¯s) as in case (1) and prove similar 

results. 

(3) Let G denote the split orthogonal group SO 2m(2n+1) .Letɛ denote a generic 

irreducible cuspidal representation of SO 2m (A), andletτ = τ(ɛ) denote its lift 

to GL 2m . We assume that τ is cuspidal. Let Q denote the standard parabolic subgroup 

of G whose Levi part is GL 2m × ··· × GL 2m × SO 2m .LetE τ(ɛ),ɛ (g, ¯s) 

denote the Eisenstein series defined on G and associated with the induced representation 

Ind G(A) 

Q(A) (τ ⊗···⊗τ ⊗ ɛ)δ¯s Q . Write an element in the Levi part of Q as 

g = diag(g 1 ,...,g n ,h,gn ∗,...,g∗ 1 ). Then we define δ¯s Q = ∏ n 

i=1 |g i| s i δ1/2 Q (g). Asin 

cases (1), (2), it follows that up to a product of local intertwining operators, the poles 

of the Eisenstein series are determined by 

∏n−1 

i=1 

L S (τ × τ,s i − s i+1 ) L S (τ × ɛ, s n ) 

L S (τ × τ,s i − s i+1 + 1) L S (τ × ɛ, s n + 1) L τ (¯s), 



the Eisenstein series has a pole at that point.



O G ( ɛ ) = ((2m) 2n (2m − 1)1). Indeed, recall that unipotent orbits for the orthogonal 

groups are parameterized by partitions such that even numbers occur with even 

multiplicity. Therefore, the Whittaker coefficients for automorphic representations of 

SO 2m (A) are attached to the unipotent orbit ((2m − 1)1). Arguing as in Theorem 1, 

the above statement regarding O G ( ɛ ) follows. 

(4) Let G denote the split orthogonal group SO (2n+1)(2m+1)+1 .Letɛ denote an 

irreducible generic cuspidal representation of Sp 2m (A), andletµ = µ(ɛ) denote its 

theta lift to SO 2(m+1) (A). Assume that ɛ lifts to a cuspidal representation τ = τ(ɛ) of 

GL 2m+1 (A). LetQ denote the standard parabolic subgroup of G whose Levi part is 

GL 2m+1 ×···×GL 2m+1 × SO 2(m+1) .LetE τ(ɛ),µ(ɛ) (g, ¯s) denote the Eisenstein series 

defined on G and associated with the induced representation Ind G(A) 

Q(A) 

(τ ⊗ ··· ⊗τ ⊗ 

µ)δ¯s 

Q . Write an element in the Levi part of Q as g = diag(g 1,...,g n ,h,gn ∗,...,g∗ 1 ). 

Then we define δ¯s Q = ∏ n 

i=1 |g i| s i δ1/2 Q (g). As in cases (1) – (3), it follows that up 

to a product of local intertwining operators, the poles of the Eisenstein series are 

determined by 

∏n−1 

i=1 

L S (τ × τ,s i − s i+1 ) L S (τ × µ(ɛ),s n ) 

L S (τ × τ,s i − s i+1 + 1) L S (τ × µ(ɛ),s n + 1) L τ (¯s), 



the Eisenstein series has a pole at that point. 


O G ( ɛ ) = ((2m + 1) 2n+1 1). 

(5) Let G denote the split orthogonal group SO 4m(n+1) .Letτ = τ(ɛ) denote a 

cuspidal representation of GL 2m (A) which is a lift from a cuspidal generic irreducible 

representation ɛ of SO 2m+1 .LetQ denote the standard parabolic subgroup of G whose 

Levi part is GL 2m ×···×GL 2m , where GL 2m occurs n + 1 times. Let E τ(ɛ) (g, ¯s) 

denote the Eisenstein series defined on G and associated with the induced representation 

Ind G(A) 

Q(A) (τ ⊗···⊗τ)δ¯s Q . Here the character δ¯s Q is defined as follows. Write 

an element in the Levi part of Q as g = diag(g 1 ,...,g n+1 ,gn+1 ∗ ,...,g∗ 1 ).Thenwe 

define δ¯s Q = ∏ n+1 

i=1 |g i| s i δ1/2 Q (g). As in the case of the Eisenstein series considered in 

Section 2.2, it is not hard to check that up to a product of local intertwining operators, 

the poles of the Eisenstein series are determined by 

n∏ L S (τ × τ,s i − s i+1 ) L S( τ, ∧2 ) 

,s n+1 

L S (τ × τ,s i − s i+1 + 1) L S( τ, ∧2 ,s n+1 + 1 )L τ (¯s), 

i=1



at the point s i = n − i + 2.Let¯s 0 = (n + 1,n+ 2,...,1). It follows from the above 

that the Eisenstein series has a pole at that point. If we denote by ɛ the residue of 

this Eisenstein series at the point ¯s 0 ,thenO G ( ɛ ) = ((2m) 2(n+1) ). 

These are the five residue representations that we consider. As mentioned above, 

one can prove analogous statements to those that we proved in detail in the beginning 

of Section 2.2. 

2.3. Definition of the lifts 

Let π denote an irreducible cuspidal generic representation defined on H (A),whereH 

is a split classical group of type B,C,orD. More specifically, in the remainder of this 

article, H denotes one of the split algebraic groups Sp 2(n+m) , SO 2(n+m)+1 , SO 2(n+m) , 

SO 2(n+m+1) or the metaplectic group ˜Sp 2(n+m) . We define the following. 

Definition 2 

We say that π is a weak endoscopic lift from two generic automorphic representations 

σ and ɛ, defined on the groups H 1 (A) and H 2 (A),ifπ is the weak functorial lift from 

σ and ɛ corresponding to the homomorphism of L-groups as given by one of the 

following cases: 

(1) if H = Sp 2(n+m) ,H 1 = Sp 2n ,andH 2 = SO 2m , then the homomorphism of 

L-groups is given by SO 2n+1 (C) × SO 2m (C) ↦→ SO 2(n+m)+1 (C); 

(2) if H = ˜Sp 2(n+m) ,H 1 = ˜Sp 2n ,andH 2 = ˜Sp 2m , then the homomorphism of 

L-groups is given by Sp 2n (C) × Sp 2m (C) ↦→ Sp 2(n+m) (C) (see Section 2.2(2)); 

(3) if H = SO 2(n+m) ,H 1 = SO 2n ,andH 2 = SO 2m , then the homomorphism of 

L-groups is given by SO 2n (C) × SO 2m (C) ↦→ SO 2(n+m) (C); 

(4) if H = SO 2(n+m+1) ,H 1 = Sp 2n ,andH 2 = Sp 2m , then the homomorphism of 

L-groups is given by SO 2n+1 (C) × SO 2m+1 (C) ↦→ SO 2(n+m+1) (C);and 

(5) if H = SO 2(n+m)+1 ,H 1 = SO 2n+1 ,andH 2 = SO 2m+1 , then the homomorphism 

of L-groups is given by Sp 2n (C) × Sp 2m (C) ↦→ Sp 2(n+m) (C). 

Our goal is to construct representations π in the above five cases. In each of the 

above cases, we introduce a group M and a representation defined on M(A). 

The representation corresponds to one of the five cases introduced at the end of 

Section 2.2. We also introduce a unipotent subgroup U of M and a character ψ U 

defined on U(F )\U(A). With this data, we construct an integral that is used to define 

the lifting. 

(a) Let M = Sp 2m(2n+1) or its double cover, where m, n are two natural numbers 

greater than or equal to 1. Let ɛ denote the automorphic representation of M(A) as 

constructed in Section 2.2(1), (2). Thus, O M ( ɛ ) = ((2m) 2n+1 ). This means that ɛ 

has no nonzero Fourier coefficient corresponding to any unipotent orbit of M which


is greater than or not related to ((2m) 2n+1 ).LetO M = ((2m − 1) 2n 1 2(m+n) ). It follows 

from [CM, Theorem 6.1.3] that the stabilizer of this orbit is the group Sp 2n ×Sp 2(n+m) . 

We associate to O M a Fourier coefficient. This association is described in detail in 

[G3]. Let P (O M ) denote the standard parabolic subgroup of M whose Levi part is 

× Sp 2(m+2n) . We denote its unipotent radical by U(O M ),orsimplybyU. If 

m = 1,wetakeU to be the trivial group. In terms of matrices, we can identify U with 

the unipotent subgroup of M given by 

GL m−1 

2n 

U = U(O M ) = U p,q,r 

⎧⎛ 

⎞⎫ 

I p x 1 ∗ 

. .. . .. ∗ 

I p x r ∗ 

I p y 1 y 2 z 

⎪⎨ 

I 

= 

q 0 y ∗ ⎪⎬ 

2 

I q y1 

∗ , (2) 

I p xr 

∗ . .. . .. 

⎜ 

⎟ 

⎝ 

I p x1 

⎪⎩ 

∗ ⎠ 

⎪⎭ 

I p 

where r = m − 2,p = 2n, andq = m + 2n. In the preceding display, x i ∈ 

Mat p×p ,y j ∈ Mat p×q ,andz ∈ Mat 0 p×p ={A ∈ Mat p×p : A t J p = J p A}. Also, the 

∗ indicates arbitrary entries such that the above matrix is in M. 

To define the character ψ U , we identify the group U/[U,U] with the additive 

group 

X = Mat 2n×2n ⊕···⊕Mat 2n×2n ⊕ Mat 2n×2(2n+m) , 

where Mat 2n×2n appear m − 2 times. 

Write an element x ∈ X as x = (x 1 ,...,x m−2 ,y), where we have x i ∈ Mat 2n×2n 

and y ∈ Mat 2n×2(2n+m) . Write y = (ȳ 1 , ȳ 2 , ȳ 3 ), where ȳ 1 , ȳ 3 ∈ Mat 2n×(n+m) and 

ȳ 2 ∈ Mat 2n×2n .Givenu ∈ U, write it as u = xu ′ , where x ∈ X and u ′ ∈ [U,U]. We 

define 

ψ U (u) = ψ U (xu ′ ) = ψ U (x) = ψ ( tr(x 1 +···+x m−2 + ȳ 2 ) ) . 

As mentioned above, the stabilizer of ψ U inside the Levi part of P (O M ) is given by 

Sp 2n × Sp 2(n+m) . The embedding is given as follows. Given that (g, h) ∈ Sp 2n × 

Sp 2(n+m) , we embed it inside GL m−1 

2n × Sp 2(m+2n) as (g, h) ↦→ (g,...,g,(g, h)). We 

use the same embedding when M is the double cover of the symplectic group.


To define the lift that we intend to study, let σ denote an irreducible cuspidal 

representation of H 1 (A), where H 1 is as in Definition 2(1), (2). In this case, we let π 

denote the automorphic representation defined on H (A) generated by the space of all 

functions 

∫ ∫ 

( ) 

f (h) = 

ϕ σ (g)θ ɛ u(g, h) ψU (u) dudg. (3) 

H 1 (F )\H 1 (A) U(F )\U(A) 

Here ϕ σ is a vector in the space of σ ,andθ ɛ is a vector in the space of ɛ . 

(b) Let M = SO 2m(2n+1) , where m, n are two natural numbers such that n ≤ m. 

Let ɛ denote the automorphic representation of M(A) which was constructed in 

Section 2.2(3). Thus, O M ( ɛ ) = ((2m) 2n (2m − 1)1).LetO M = ((2m − 1) 2n 1 2(m+n) ). 

It follows from [CM, Theorem 6.1.3] that the stabilizer of this orbit is the group 

SO 2n × SO 2(n+m) .LetP (O M ) denote the standard parabolic subgroup of M whose 

Levi part is GL m−1 

2n × SO 2(m+2n) . We denote its unipotent radical by U(O M ),orsimply 

by U.Ifm = 1,wetakeU to be the trivial group. In term of matrices, we can identify 

U with the unipotent subgroup of M givenby(2) sothatr = m − 2,p = 2n, and 

q = m + 2n and so that the following conditions are satisfied. First, z ∈ Mat 0 p×p = 

{A ∈ Mat p×p : A t J p =−J p A}. Second, the ∗ indicates that the entries are such 

that U is a subgroup of SO 2m(2n+1) . 

Since we can identify U/[U,U] with the group X as defined in case (a), we 

define the character ψ U as in that case. The stabilizer of ψ U is SO 2n × SO 2(n+m) ,and 

it is embedded inside M in a way similar to the embedding of the stabilizer in case 

(a). The definition of the representation π which we construct is given by the space 

generated by the functions 

∫ ∫ 

( ) 

f (h) = 

ϕ σ (g)θ ɛ u(g, h) ψU (u) dudg. 

H 1 (F )\H 1 (A) U(F )\U(A) 

Here σ is a cuspidal representation of the group H 1 = SO 2n ,andθ ɛ is a vector in the 

space of ɛ . The group U and the character ψ U are defined above. 

(c) Let M = SO (2m+1)(2n+1)+1 , and assume that m ≥ n. Let ɛ denote the 

automorphic representation of M(A), as was constructed in Section 2.2(4). Thus, 

O M ( ɛ ) = ((2m + 1) 2n+1 1). LetP (O) denote the standard parabolic subgroup 

of M whose Levi part is GL m 2n × SO 2(n+m+1). We denote its unipotent radical by 

U = U(O). In term of matrices, we write this unipotent group as in (2), where 

r = m − 1,p = 2n, andq = m + n + 1. In this case, x i ∈ Mat 2n×2n ,y 1 ,y 2 ∈ 

Mat 2n×(m+n+1) ,andz ∈ Mat 0 2n×2n = {A ∈ Mat 2n×2n : A t J 2n = −J 2n A}. We 

define ψ U (u) = ψ(tr(x 1 + ··· + x l−1 )). LetH 4n(m+n+1)+1 denote the Heisenberg 

group with 4n(m + n + 1) + 1 variables. From the definition of U, it follows 

that there is a projection l from U onto the Heisenberg group H 4n(m+n+1)+1


defined as follows. Identify elements of H 4n(m+n+1)+1 with triples (x,y,z ′ ),where 

x,y ∈ Mat 2n×(m+n+1) and z ′ ∈ Mat 1×1 .Givenu ∈ U in the above coordinates, the 

projection is given by l(u) = (y 1 ,y 2 , tr ′ z). Here, for z = (z i,j ) ∈ Mat 0 2n×2n ,wedefine 

tr ′ z = z 1,1 +···+z n,n . The stabilizer of the character ψ U is Sp 2n × SO 2(n+m+1) .It 

is embedded inside GL m 2n × SO 2(n+m+1) as (g, h) ↦→ (g, g, . . . , g, h). Letσ denote 

a cuspidal representation of H 1 (A) = Sp 2n (A). We define the representation π of 

H (A) = SO 2(n+m+1) (A) as the space generated by all functions 

∫ ∫ 

f (h) = 

ϕ σ (g)θ ψ ( ) ( ) 

Sp 4n(m+n+1) 

l(u)(g, h) θɛ u(g, h) ψU (u) dudg. 

H 1 (F )\H 1 (A) U(F )\U(A) 

(4) 

Here θ ψ Sp 4n(m+n+1) 

is the theta function defined on H 4n(m+n+1)+1 · ˜Sp 4n(m+n+1) (for basic 

definitions regarding the theta representation, see [P]). Also, ϕ σ is a vector in the space 

of σ ,andθ ɛ is a vector in the space of ɛ . The embedding of Sp 2n ×SO 2(m+n+1) inside 

Sp 4n(m+n+1) is given by the tensor product. 

(d) To describe the last case in Definition 2, letM = SO 4m(n+1) , and, as before, 

assume that m ≥ n. Let ɛ denote the automorphic representation of M(A) which 

was defined in Section 2.2(5). Thus O M ( ɛ ) = ((2m) 2(n+1) ).LetO M = ((2m − 

1) 2n+1 1 2(m+n)+1 ).LetP (O) denote the parabolic subgroup of M whose Levi part is 

GL m−1 

2n+1 × SO 2(m+2n+1). We denote by U = U(O) the unipotent radical subgroup of 

P (O). We use the matrix description given in formula (2), adopted as in (b) to the 

even orthogonal group, with r = m − 2,p = 2n + 1, andq = m + 2n + 1. Thus 

U/[U,U] can be identified with the group 

X = Mat (2n+1)×(2n+1) ⊕···⊕Mat (2n+1)×(2n+1) ⊕ Mat (2n+1)×2(m+2n+1) , 

where Mat (2n+1)×(2n+1) appears m − 2 times. Write an element x ∈ X as x = 

(x 1 ,...,x m−2 ,y), where x i ∈ Mat (2n+1)×(2n+1) and y ∈ Mat (2n+1)×2(2n+m+1) . 

Given u ∈ U, write it as u = xu ′ , where x ∈ X and u ′ ∈ [U,U]. Wedefine 

ψ U (u) = ψ U (xu ′ ) = ψ U (x) = ψ ( tr(x 1 +···+x m−2 ) + tr ′ y) ) . 

Here ψ(tr ′ y) = ψ(y 1,1 +···+y n,n +y n+1,m+2n+1 +y n+1,m+2n+2 +y n+2,2(m+2n+1)−n+1 + 

···+y 2n+1,2(m+2n+1) ). One can verify that the stabilizer of the character ψ U is the group 

SO 2n+1 × SO 2(n+m)+1 and that its embedding is similar to that of the corresponding 

groups in the previous cases. 

To define π, we start with a cuspidal representation σ of H 1 (A) = SO 2n+1 (A), 

and we use a similar integral representation as defined in formula (3). 

3. The cuspidality of the lift 

We continue with the notation of Section 2. Let ɛ denote an automorphic representation 

of the group M(A), as constructed in Section 2. In this section, we discuss the


cuspidality of the lift. Since the computations are quite similar in all five cases, we 

concentrate only on the first case. 

Let M = Sp 2m(2n+1) .Letσ denote an irreducible cuspidal representation of 

Sp 2n (A). In this case, the lift is given by the integrals 

∫ ∫ 

( ) 

f (h) = 

ϕ σ (g)θ ɛ u(g, h) ψU (u) dudg, 

H 1 (F )\H 1 (A) U(F )\U(A) 

where H 1 = Sp 2n and h ∈ H = Sp 2(m+n) . (The group U and the character ψ U were 

described in Section 2.) 

As often happens in the constructions of liftings using small representations, the 

image of the lift is not always cuspidal. Usually, there is an obstruction for the lift to 

be cuspidal. To understand when this can happen, assume, for example, that n ≥ m. 

Assume that σ itself is an endoscopic lift from two cuspidal representations, from a 

cuspidal representation σ ′ on Sp 2(n−m) and from a cuspidal representation ɛ ′ on SO 2m . 

For example, if ɛ ′ = ɛ, then it is not expected that the representation π is cuspidal. In 

this case, there is an obstruction for the lift to be cuspidal, which is basically expressed 

in terms of lifts to groups of smaller rank. 

To simplify notation, we prove Theorem 2 only when n ≤ m. Whenn>m, 

the formal computations of the constant terms are similar. Since we assumed that the 

cuspidal representation ɛ lifts to a cuspidal representation τ(ɛ) on GL 2m , the image 

of the lift can fail to be cuspidal only in the case where n = m. Thus, in this case, for 

the image to be cuspidal we have to assume that a certain integral is zero. In our case, 

the integral we need to assume to be zero is given by integral (7), defined in the proof 

of Theorem 2. One can interpret this integral as a lift to a group of a lower rank. 

The proof of the cuspidality of the lift requires a manipulation of Fourier expansions 

performed on the automorphic functions θ ɛ . Here the function θ ɛ liesinthe 

space of the residue representation ɛ . At each step, one has to check that the integrals 

converge absolutely. These justifications are now quite standard; the main reference 

required is [MW, I.2.10]. 

THEOREM 2 

With the above notation, let π denote the automorphic representation of Sp 2(m+n) (A) 

generated by the space of functions f (h) defined above. Assume that n ≤ m. Inthe 

case where n = m, assume also that the integral (7) is zero for every choice of data. 

Then π is a cuspidal representation. 

Proof 

For 1 ≤ j ≤ m + n, letV j denote the standard unipotent radical of the maximal 

parabolic subgroup of H whose Levi part is GL j × Sp 2(m+n−j) . Thus, we prove that


f V j 

(h) is zero for every choice of data. In other words, we need to consider the 

integrals 

∫ ∫ ∫ 

( ) 

ϕ σ (g)θ ɛ u(g, v) ψU (u) dv dudg. (5) 

H 1 (F )\H 1 (A) U(F )\U(A) V j (F )\V j (A) 

The group V j is embedded inside H as the group of all matrices of the form 

⎧⎛ 

⎞ 

⎫ 

⎨ I j X ′ Y ′ 

⎬ 

V j = ⎝ I 

⎩ 2(m+n−j) X ′ ∗⎠ : X ′ ∈ Mat j×2(m+n−j) ,Y ′ ∈ Mat 0 j×j⎭ . 

I j 

Let w denote the Weyl element of M defined by 

⎛ 

I j 

I k1 w = 

⎜ I 2(m+2n−j) ⎟ 

⎝ 

I k1 

⎠ , k 1 = 2n(m − 1). 

I j 

Conjugating in integral (5) the argument of the function θ ɛ by w, and using the 

left-invariance property of this function by rational points, we obtain 

∫ 

ϕ σ (g)θ ɛ 

( 

t(Z, Y, R)u(g, 1) w t(X)w ) ψ m,n,j (u) dudY dZ dR dXdg. 

⎞ 

Here 

⎛ 

I j Z Y 1 Y 2 R 

⎞ 

I k1 Y ∗ ⎛ 

2 

I j 

I k2 Y ∗ 1 

X I k1 

t(Z, Y, R) = 

I 2n , t(X) = 

I k3 I k2 Z ∗ 

⎜ 

⎝ 

I k1 

⎜ 

⎟ 

⎝ 

I k1 ⎠ 

X ∗ 

I j 

⎞ 

, 

⎟ 

⎠ 

I j 

where k 1 = 2n(m − 1),k 2 = m + n − j,andk 3 = 2(m + 2n − j). Here Y = (Y 1 ,Y 2 ), 

and all matrices are such that the above two matrices are in M. In the above integral, 

these matrices are integrated over Mat r1 ×r 2 

(F )\Mat r1 ×r 2 

(A) with the appropriate 

values of r i . Next, we integrate u ∈ U 2n,m+2n−j,m−2 (F )\U 2n,m+2n−j,m−2 (A), 

where the group U p,q,r wasdefinedin(2). The unipotent group U 2n,m+2n−j,m−2 

is a subgroup of Sp 2(2mn+m−j) . We view it as a subgroup of M, by embedding 

it as all matrices of the form diag(I j ,u,I j ). For the group U p,q,r , we defined


immediately following (2) a character of ψ Up,q,r which was denoted there by ψ U . 

In the above integral, we write ψ m,n,j for the character ψ U2n,m+2n−j,m−2 . Finally, we have 

(g, 1) w = w(g, 1)w −1 = diag(I j ,g,...,g,I m+n−j ,g,I m+n−j ,g,...,g,I j ), where 

the dots indicate that g occurs m − 1 times. 

Let α(S) = α(S 1 ,S 2 ) denote the unipotent subgroup of M defined by 

⎛ 

α(S 1 ,S 2 ) = 

⎜ 

⎝ 

I j 

⎞ 

S 

I 2n I k4 S ∗ 

⎟ 

I 2n 

⎠ , 

I j 

k 4 = 2m(2n + 1) − 2(2n + j). 

Here S = (S 1 0 k2 S 2 0 k1 +k 2 −2n) ∈ Mat j×k4 , where S 1 ∈ Mat j×2n(m−2) ,S 2 ∈ Mat j×2n , 

and 0 p represents a zero matrix of size j ×p. We now expand the above integral along 

the group α(S), where S is integrated over points in (A) modulo points in F .Wehave 

∫ ∑ 

∫ 

( 

ϕ σ (g)θ ɛ α(S) t(Z, Y, R)u(g, 1) w t(X)w ) 

δ 

× ψ m,n,j (u)ψ δ (S) dS dudY dZ dR dXdg, 

where δ is summed over all characters of the group S(F )\S(A). We can identify this 

group of characters with all matrices Mat j×2n(m−1) (F ).Sinceθ ɛ is left-invariant under 

rational points, it is left-invariant under all matrices {t(δ) : δ ∈ Mat 2n(m−1)×j }.We 

conjugate the matrix t(δ) across α(S) t(Z, Y, R) u(g, 1) w from left to right. It follows 

from a matrix multiplication that after we change variables in u, the character ψ δ (S) 

is canceled. Thus we obtain, after conjugation, the matrix t(X + δ), and we can then 

collapse the summation over δ with the integration over X. We obtain 

∫ 


( 

u1 u(g, 1) w t(X)w ) ψ m,n,j (u) du 1 dudXdg. (6) 

Here u 1 ∈ U 1 , which is defined as the unipotent group of M generated by all matrices 

of the form 

⎛ 

I j 

˜Z Ỹ 

u 1 = ⎜ 

⎝ I k5 

˜Z ∗ ⎟ 

⎠ , k 5 = 2(2mn + m − j), 

I j 

⎞ 

where ˜Z = (0 Z 1 ) ∈ Mat j×2(2mn+m−j) with Z 1 ∈ Mat j×2(2mn+m−n−j) and Ỹ is such that 

the group U 1 is in M. Also, the matrix t(X) is integrated over X ∈ Mat 2n(m−1)×j (A). 

The variable u is integrated as before.


Next, we expand the above integral along the group of all matrices of the form 

⎛ 

β(R) = 

⎜ 

⎝ 

I j 

R 

I 2n 

I k4 

I 2n R ∗ 

I j 

⎞ 

⎟ 

⎠ , 

R ∈ Mat j×2n. 

First, we consider the contribution to the Fourier expansion from the constant term. 

For 1 ≤ j ≤ m + n, letU j (M) denote the unipotent radical of the standard maximal 

parabolic subgroup of M whose Levi part is GL j × Sp 2(2mn+m−j) . Since the group 

generated by U 1 and by {β(R) :R ∈ Mat j×2n } equals U j (M), it follows that we 

obtain the constant term θ U j (M) 

ɛ as an inner integration. From the definition of the 

representation ɛ and from the fact that j ≤ m + n ≤ 2m, it follows that if n


expansion. Here the embedding of this group inside Sp 2m(2n+1) (F ) is 

(k, g) ↦→ diag(k,g,...,g,I m+n−j ,g,I m+n−j ,g,...,g,k ∗ ). 

Since χ is nontrivial, we may assume after a conjugation by a suitable element of 

GL j (F ) × Sp 2n (F ) that for R = (R i,j ),wehaveχ(R) = ψ(R 1,1 +···). Assume first 

that χ is such that χ(R) = ψ(R 1,1 + R j,2n +···), where the dots indicate that χ is 

trivial on all entries R 1,l for l>1 and also trivial on R l,2n with l


As with the group T 0 , we suppress j from the notation. Notice that S 0 is, in fact, a 

subgroup of the group of matrices of the form β(R). The function θ ɛ is left-invariant 

under rational points. Thus, given a character ν as above, one can find an element 

s 0 (ν) ∈ S 0 (F ) such that when we conjugate the above integral by that element from 

left to right, we can collapse summation and integration in such a way that we obtain 

the integral 

∫ 

θ ɛ 

( 

t0 β(R)u 1 u ) ψ m,n,j (u)χ(R) dt 0 dR du 1 du 

as inner integration to the above integral. Here the variable R is integrated over the 

group S 0 (A)B ′ (F )\B ′ (A), where B ′ ={β(R) :R ∈ Mat j×2n }. Thus, to prove that 

(10) is zero, it is enough to prove that the above integral is zero. 

We proceed by induction. For 1 ≤ l ≤ m − 1, we define the following 2m − 1 

families of abelian unipotent subgroups of Sp 2m(2n+1) . Here a = 2(mn + m + n): 

T l = 

S l = 

{ 2nl+j 

∑ 

I + 

k=1 

{ 2nl+j 

∑ 

I + 

k=1 

} 

r k e ′ 2n(l−1)+j+1,k : r 1 = r 2ns+j+1 = 0; 0 ≤ s ≤ l − 1 , 

} 

p k e ′ k,2nl+j+1 : p 1 = p 2ns+j+1 = 0; 0 ≤ s ≤ l − 1 , 

{ a−j 

∑ 

} 

T m = I+ r k e ′ 2mn+m−n+1,k : r 1 = r 2ns+j+1 = r 2mn+m−n+1 = 0; 0 ≤ s ≤ l−1 , 

k=1 

{ a−j 

∑ 

} 

S m = I + p k e ′ k,a−j+1 : p 1 = p 2ns+j+1 = p 2mn+m−n+1 = 0; 0 ≤ s ≤ l − 1 , 

k=1 

{ a−j+2nl 

∑ 

T m+l = I + r k e ′ a+2n(l−1)−j+1,k : r 1 = r 2ns+j+1 = 0; 

k=1 

} 

r 2mn+m−n+1 = r a+2nq−j+1 = 0; 0 ≤ s ≤ m − 2; 0 ≤ q ≤ l − 1 , 

{ a−j+2nl 

∑ 

S m+l = I + p k e ′ k,a+2nl−j+1 : p 1 = p 2ns+j+1 = 0; 

k=1 

} 

p 2mn+m−n+1 = p a+2nq−j+1 = 0; 0 ≤ s ≤ m − 2; 0 ≤ q ≤ l − 1 . 

All of the above groups depend on the parameter j, which we omit from the notation.


Notice that S l are all subgroups of the group U 1 U m,n,j defined above. Recall that 

in the above integral, we integrate along these two groups. We inductively expend the 

above integral along these groups. More precisely, we start by expending the above 

integral along T 1 (F )\T 1 (A). Then by arguing as above with the groups T 0 and S 0 ,we 

use S 1 to collapse summation and integration. Next, we proceed similarly with the 

pair T 2 and S 2 and so on. Performing this process 2m times, we obtain the integral 

∫ 

θ ɛ 

( 

t0 t 1 ···t 2m−1 β(R)u 1 u ) ψ m,n,j (u)χ(R) dt l dR du 1 du. 

Here the variables t l are integrated over T l (F )\T l (A), and the variables β(R)u 1 u are 

integrated over 

S 0 (A) ···S 2m−1 (A)β(R)(F )U 1 (F )U m,n,j (F )\β(R)(A)U 1 (A)U m,n,j (A). 

Let w 0 denote the following Weyl element of Sp 2m(2n+1) .Letw 0 [i, k] denote the (i, k)- 

entry of w 0 .Wesetw 0 [1, 2] = w 0 [l,2n(l−2)+j+1] = w 0 [m+1, 2mn+m−n+1] = 

w 0 [m + l,2(mn + m + n) + 2n(l − 2) − j + 1] = 1, where 2 ≤ l ≤ m. All other 

entries of the first 2m rows are zero. For w 0 to be symplectic, this determines the last 

2m rows of w 0 uniquely. At the rest of the rows, we choose the entries of w 0 to be 

such that w 0 is a monomial matrix in Sp 2m(2n+1) such that all nonzero entries are 1 or 

−1. Clearly, θ ɛ is left-invariant by w 0 . Using that, if we conjugate from left to right 

by w 0 , then it is not hard to check that we obtain integral (9) with l = 2m as inner 

integration. As explained above, this integral is zero. This proves that the contribution 

to (8) of summands that correspond to characters of type A is zero. 

The second type of characters are those not of type A and such that the stabilizer 

inside GL j (F ) contains the unipotent radical of a parabolic subgroup of GL j .This 

can happen in the following situation. If χ is not of type A, then we can choose it 

to be as follows. Write R = (R i,k ). Then there exists a number l < j such that 

χ(R) = ψ(R l,1 + R l+1,2 +···+R j,j−l−1 ). We refer to such characters as characters 

of type B. For these characters, we can further expend their contribution to integral 

(8). In this case, we get the sum 

∑ 

∫ 

( 

θ ɛ β ′ (P )β(R)u 1 u ) ψ m,n,j (u)χ(R)µ(P ) dP dR du 1 du. (11) 

µ 

Here 

⎛ 

β ′ (P ) = 

⎜ 

⎝ 

I j−l 

⎞ 

P 

I l I b ⎟ 

I l P ∗ ⎠ , b = 2m(2n + 1) − 2j, 

I j−l


and µ is summed over all characters of this group. Now, we argue as in the case of 

characters χ as above. More precisely, if µ is trivial, we argue as we did right before 

equation (8) with the case of the trivial orbit, and we show that it has zero contribution 

to (11). If µ is nontrivial, then we can choose it to be of the form µ(P ) = µ(P 1,1 +···), 

where the dots indicate that µ is trivial on all entries P 1,k , where k>1. Now, we argue 

as we did with characters χ of type A. In other words, we define suitable groups T i 

and S i as above and show that we obtain integral (9) with l = m as inner integration. 

Thus, we obtain zero contribution in (8) also from characters χ of type B. 

The last type of characters that we need to handle in (8) are those not of type A and 

for which the stabilizer in GL j does not contain a unipotent radical. This can happen 

only if j


values of m ′ and n ′ . The variable m(Y ) is integrated over L(A), where L is a certain 

unipotent group. Finally, the matrix w 0 is a suitable Weyl element. 

The key point here is that when we replace the variable g in (14) byug, where 

u ∈ U j (Sp 2n )(A), we can conjugate it to the left. Using the integration over the 

unipotent group V 2m(2n+1),2mj (M)(A) by changing variables, we obtain the fact that 

(14) is left-invariant by the group U j (Sp 2n )(A). As explained above, this implies that 

the type-C characters also contribute zero to integral (8). 

Combining all of this completes the proof of the cuspidality. 

 

4. The nonvanishing of the lift 

In this section, we prove that the lift constructed in Section 2.3 is nonzero. We do this 

by computing the Whittaker coefficient of the lift and showing that it is not zero. In 

particular, this proves that the image of the lift contains cuspidal representations that 

are generic. As before, we give the details of one case. (The other cases are similar.) 

These types of computations are now quite familiar; there are many examples of them 

in the literature (see, e.g., [GJ], [GRS4]). Therefore, we indicate the necessary steps 

of these computations, in some places sketchily. 

We consider the first case introduced in Definition 2. In that case, the lift is given 

in terms of integral (3), where H 1 = Sp 2n . The group U and the character ψ U are 

described there explicitly. Let V (Sp 2k ) denote the maximal standard unipotent radical 

for the group Sp 2k . This group consists of upper unipotent matrices. Let ψ V (Sp2k ),a 

denote the Whittaker character defined on the group V (Sp 2k ). In more detail, if 

v = (v i,j ) ∈ V (Sp 2k ), then we define ψ V (Sp2k ),a = ψ(v 1,2 +···+v k−1,k + av k,k+1 ), 

where a ∈ (F ∗ ) 2 \F ∗ . 

Recall from Section 2 that the definition of the representation ɛ depends on a 

generic automorphic representation µ(ɛ) of Sp 2m . Thus, there exists an a ∈ F ∗ such 

that µ(ɛ) has a nonzero Whittaker-Fourier coefficient ψ V (Sp2m ),a. Using the notation of 

(3), our goal in this section is to compute the integral 

∫ 

f (vh)ψ V (Sp2(m+n) ),a(v) dv. (15) 

V (Sp 2(m+n) )(F )\V (Sp 2(m+n) )(A) 

As with the cuspidality condition studied in Section 3, to prove the nonvanishing 

of (15) requires that we perform a certain quantity of Fourier expansions. The 

convergence of each of the integrals is justified using [MW, I.2.10]. 

We start by introducing certain notation. For 1 ≤ i ≤ m 2 ,letMat col,i 

m 1 ×m 2 

denote 

the group of all matrices whose last i columns are zero. Similarly, for 1 ≤ i ≤ m 1 ,let 

Mat row,i 

m 1 ×m 2 

denote the group of all matrices whose last i rows are zero.


by 

Let ̂R 1 denote the group of all unipotent matrices inside V(Sp 2m(2n+1) ) defined 

⎧⎛ 

⎪⎨ 

̂R 1 = 

⎜ 

⎝ 

⎪⎩ 

I 2n(m−1) 

⎞ 

R 

⎛ 

I n+m I 2n ⎟ 

I n+m R ∗ ⎠ ,R= ⎜ 

⎝ 

I 2n(m−1) 

R m−1 

R m−2 

. 

R 1 

⎞ 

⎟ 

⎠ : R i ∈ Mat col,i 

2n×(n+m) 

Also, let S denote the group of all unipotent matrices inside Sp 2m(2n+1) defined by 

⎧⎛ 

⎪⎨ 

S = 

⎜ 

⎝ 

⎪⎩ 

I 2n(m−1) 

S 2 I n+m S 1 S 3 

I 2n S1 

∗ 

I n+m 

S2 

∗ 

⎫ 

⎪⎬ 

. 

⎞ 

⎫ 

⎟ 

⎠ ,S 2 = ( ) 

⎪⎬ 

0 S 2,m−1 ··· S 2,3 S 2,2 . 

⎪⎭ 

I 2n(m−1) 

Here S 1 ∈ Mat row,1 

(n+m)×2n ,S 2,i ∈ Mat row,i 

(n+m)×2n ,andS 3 ∈ Mat 0 (n+m)×(n+m) 

are such that the 

first column of S 3 is zero. The center of the group S, which we denote by Ŝ, consists of 

all matrices in S such that S 1 and S 2 are zero. Thus, Ŝ is an abelian unipotent subgroup 

of V (Sp 2m(2n+1) ). 

Returning to integral (15) as defined by integral (3), we start by expanding it along 

the group Ŝ(A)S(F )\S(A). We obtain 

∑ 

∫ 

( ) 

ϕ σ (g)θ ɛ su(g, vh) ψU (u)ψ V (Sp2(m+n) ),a(v)ψ α (s) ds dv dudg. 

α 

Here g is integrated over Sp 2n (F )\Sp 2n (A), the variable u is integrated over the 

group U(F )\U(A) (see (3)), the variable s is integrated over Ŝ(A)S(F )\S(A), 

and v is integrated as in (15). The variable α is summed over all characters of 

the group Ŝ(A)S(F )\S(A). Notice that ̂R 1 is a subgroup of U. Sinceθ ɛ is leftinvariant 

over rational points, conjugating in the above integral by a suitable rational 

matrix in ̂R 1 (F ), and after a suitable collapsing of summation and integration, we 

obtain 

∫ 

( 

ϕ σ (g)θ ɛ su(1,v)̂r1 (g, h) ) ψ U (u)ψ V (Sp2(m+n) ),a(v) ds dudv d̂r 1 dg. 

⎪⎭ 

Here the variable u is integrated over U(F )̂R 1 (A)\U(A), and the variable ̂r 1 is integrated 

over ̂R 1 (A). All other variables are integrated as before.


For 1 ≤ i ≤ m − 1, denote by ν i the Weyl element of Sp 2m(2n+1) : 

⎛ 

ν i = 

⎜ 

⎝ 

I 2n(m−i−1) 

I m+n−i 

I 2n 

I 2(2ni+i−n) 

I 2n 

I m+n−i 

⎞ 

. 

⎟ 

⎠ 

I 2n(m−i−1) 

Denote ˜w 1 = ν m−1 ν m−2 ···ν 1 .Since˜w 1 is an element in Sp 2m(2n+1) (F ), the function 

θ ɛ (z) is left-invariant under this element. Thus θ ɛ (su(1,v)z) = θ ɛ (u ′˜w 1 z), where 

u ′ = ˜w 1 su(1,v)˜w −1 

1 . For a natural number p, letU p denote the standard unipotent 

radical of the standard parabolic subgroup of Sp 2m(2p+1) whose Levi part is GL 3p+1 × 

GL m−2 

2p+1 × Sp 2p+2 . Thus, the above integral is equal to 

∫ 

( 

ϕ σ (g)θ ɛ (v1 ,v 2 ) ′ n u n˜w 1̂r 1 (g, h) ) ( 

ψ n (v1 ,v 2 ) ′ n ,u n) 

dvi du n d̂r 1 dg. 

Here u n ∈ U n ,and 

⎛ 

(v 1 ,v 2 ) ′ n = ⎜ 

⎝ 

⎞ 

⎛ ⎞ 

1 x 

⎟ 

⎠ , v 1 ∈ V (GL n+1 ),v 2 = ⎝ I 2n 

⎠, x∈ A, 

1 

v 1 

I q 

v 2 

I q 

v ∗ 1 

(16) 

where q = (m − 2)(2n + 1) + 2n. To describe ψ n , it is convenient, for reasons that 

become clear later, to describe ψ p for any natural number p. 

To do so, write v 1 = (v 1 (i, j)) ∈ V (GL p+1 ). Then for v 2 as above, the restriction 

of ψ p to (v 1 ,v 2 ) ′ p is given by ψ p((v 1 ,v 2 ) ′ p ) = ψ(v 1(1, 2) +···+v 1 (n, n + 1) + x). 

Next, identify the group U p modulo its commutator with the group of all matrices 

(X 1 ,X 2 ,...,X m−1 ), where X 1 ∈ Mat (3p+1)×(2p+1) ;for2 ≤ i ≤ m − 2, wehave 

X i ∈ Mat (2p+1)×(2p+1) and X m−1 ∈ Mat (2p+1)×(2p+2) . Write 

X 1 = 

( ) 

X1,1 

, 

X 1,2 

where X 1,1 ∈ Mat p×(2p+1) and X 1,2 ∈ Mat (2p+1)×(2p+1) . Also, write X m−1 = 

(X m−1,1 X m−1,2 ), where X m−1,1 ∈ Mat (2p+1)×(2p+1) and X m−1,2 ∈ Mat (2p+1)×1 .Ifwe 

identify an element u p ∈ U p /[U p ,U p ] with (X 1 ,X 2 ,...,X m−1 ), then the restriction 

of ψ p to U p is given by ψ Up (u p ) = ψ(tr(X 1,2 + X 2 +···+X m−2 + X m−1,1 )).


For 1 ≤ j ≤ n, letL j ={I 2m(2n+1) + l j,1 e j,n+2 ′ + ··· + l j,2ne j,3n+1 ′ },where 

e 

p,q ′ = e p,q − e 2m(2n+1)−q+1,2m(2n+1)−p+1 and e p,q is the (2m(2n + 1) × 2m(2n + 1))- 

matrix with 1 at the (p, q)-position and zero elsewhere. Thus, we can identify the 

group L j with Mat 1×2n . Expand the above integral along the group L 1 (F )\L 1 (A). 

From the embedding of Sp 2n inside Sp 2m(2n+1) , we deduce that Sp 2n (F ) actsonthe 

character group of L 1 (F )\L 1 (A) with two orbits, one trivial and the other nontrivial. 

Thus, the expansion of the above integral breaks into a sum of two terms: first, the 

one corresponding to the nontrivial orbit and which we denote by I 1 ; then, the one 

corresponding to the trivial orbit and which we denote by I 2 .InI 2 , we expand along 

the group L 2 (F )\L 2 (A). Once again, this expansion breaks into a sum of two terms 

according to the action of Sp 2n (F ). Continuing this process, we obtain that the above 

integral can be written as a finite sum of terms corresponding to the above Fourier 

expansions. Below, we compute I 1 . Proceeding in a similar way, one obtains the fact 

that all other terms contribute zero to the expansion. Indeed, this happens since we 

either obtain constant terms that ɛ does not support, or we obtain Fourier coefficients 

of θ ɛ corresponding to unipotent orbits greater than or not related to ((2m) 2n+1 ).By 

adopting Theorem 1 to this case, these Fourier coefficients are zero. 

Thus, the above integral equals I 1 , which is equal to 

∫ 

( 

ϕ σ (g)θ ɛ (v1 ,v 2 ) ′ n l 1u n˜w 1̂r 1 (g, h) ) ( 

ψ n (v1 ,v 2 ) ′ n ,u n) 

ψ(l1,1 ) dv i dl 1 du n d̂r 1 dg, 

where we identified l 1 ∈ L 1 with (l 1,1 ,...,l 1,2n ). All variables are integrated 

as in the previous integral, except the variable g, which is integrated over 

Sp 2(n−1) (F )Z n (F )\Sp 2n (A). Here Z n is the standard unipotent radical of the maximal 

parabolic subgroup of Sp 2n whose Levi part is GL 1 × Sp 2(n−1) . Indeed, the group 

Sp 2(n−1) (F )Z n (F ) is the stabilizer inside Sp 2n (F ) of the nontrivial orbit in the above 

expansion. 

If n>1, letx β1 (1) = I 2m(2n+1) − e ′ 2,n+2 .Whenn = 1, letx β 1 

(1) = I 6m − 

∑ m 

i=1 e′ 3i−1,3i . The function θ ɛ is left-invariant under x β1 (1). Hence we can conjugate 

it from left to right. After a suitable change of variables in l 1 , we obtain 

∫ 


( 

(v1 ,v 2 ) ′ n l 1u n x β1 (1)˜w 1̂r 1 (g, h) )˜ψ n 

( 

(v1 ,v 2 ) ′ n ,u n,l 1 

) 

dvi dl 1 du n d̂r 1 dg. 

Here ˜ψ n is defined as follows. First, the restriction to v 2 and to u n is defined as 

in ψ n .Onl 1 , we define it to be ψ(l 1,1 ), and the restriction to v 1 = (v 1 (i, j)) is 

ψ(v 1 (2, 3) +···+v 1 (n, n + 1)). 

Recall that for the symplectic group, one can choose representatives of Weyl 

elements to consist of permutation matrices having 1’s and −1’s. Let ˜w 2 (n) denote 

the following Weyl element in Sp 2m(2n+1) . We write it as a permutation matrix, as 

above, and we indicate for which entries it is nonzero (i.e., ±1). First, for the first


2m rows, we have a nonzero entry at the (1, 1)-position and for 2 ≤ i ≤ 2m at the 

(i, (2i − 3)n + i)-positions. Next, in the last 2m rows, we have a nonzero entry at the 

(4mn + i, (2i + 1)n + i)-position for all 1 ≤ i ≤ 2m − 1 and a nonzero entry at the 

(2m(2n + 1), 2m(2n + 1))-entry. Finally, the rows between the 2m + 1 row and 

the 4mn row form a matrix of size 2m(2n − 1) × 2m(2n + 1) given by the matrix 

⎛ 

⎞ 

0 I n 0 

0 I 

M 

I 

M 

. .. . 

I 

⎜ 

M 

⎟ 

⎝ 

I 0 ⎠ 

0 I n 0 

Here the zero represents a column of zeros, I is the identity matrix of size 2n − 2,and 

M is the (1 × 3)-matrix defined by M = (010). In the above integral, we conjugate 

by ŵ 2 (n) and obtain 

∫ (( )( )( Z Y X I I2m 

ϕ σ (g)θ ɛ I q Y ∗ A I q 

Z ∗ B A ∗ I 

(v 1 ,v 2 ) ′ n−1 u n−1 

) 

I 2m 

) 

× ˜w 2 (n)x β1 (1)˜w 1˜r 1 (g, h) 

( 

ψ n−1 (v1 ,v 2 ) ′ n−1 ,u n−1) 

ψV (GL2m)(Z) d (···). (17) 

Here Z ∈ V (GL 2m ), the standard maximal unipotent subgroup of GL 2m .Thevariable 

X is integrated over the group Mat 0 2m×2m with the condition that X i,j = 0 if i>j. 

The variable B is integrated over the group ̂B, which is defined as the subgroup of 

Mat 0 2m×2m with the condition that B i,j = 0 if i>jand also that B i,i+1 = 0 for all 

1 ≤ i ≤ 2m − 1.Letq = 2m(2n − 1). The variable A is integrated over the subgroup 

Â of Mat 2m(2n−1)×2m defined as follows. Write 

A = 

( 

A1 

A 2 

) 

, 

where A 1 ,A 2 ∈ Mat m(2n−1)×2m . Then we integrate over all A 1 with the condition that 

the first two columns are zero and that the (i, j)-entry is zero for all 3 ≤ j ≤ m + 1 

and i>(2j − 3)n − j + 1. The matrix A 2 is integrated over all matrices such that 

all m + 1 rows are zero and the (j,2m − i)-entry is zero for all 0 ≤ i ≤ m − 2 and 

j ≥ (2i + 3)n − (i + 2). Finally, the variable Y is integrated over the subgroup Ŷ of


Mat 2m×2m(2n−1) , defined as follows. Denote by Mat ′ 1×2m(2n−1) 

the row vectors such that 

all entries except the (1, 2m(2n − 1))-entry are zero. Recall that the first two columns 

of Â are zero. Thus, by ignoring the first two columns, we can identify Â with a 

subgroup Â ′ of Mat 2m(2n−1)×(2m−2) . With this notation, the variable Y is an element in 

the group 

⎧ ⎛ ⎞ 

⎫ 

⎨ Y 1 

⎬ 

Ŷ = 

⎩ Y = ⎝Y 2 

⎠,Y 1 ∈ Mat 1×2m(2n−1) ,Y 2 ∈ J 2m(2n−1) Â ′ J 2m−2 ,Y 3 ∈ Mat ′ 1×2m(2n−1) 

⎭ . 

Y 3 

Finally, the variables (v 1 ,v 2 ) ′ n−1 vary over the group of matrices as defined in (16), 

replacing n by n − 1, and the variable u n−1 ∈ U n−1 , a group defined right before (16). 

In integral (17), all the variables described so far are integrated over their groups of 

definition with points in A modulo points in F . The variables˜r 1 and g are integrated 

as before. 

Denote by ̂R 2 the subgroup of Sp 2m(2n+1) which consists of all matrices 

⎧⎛ 

⎞ 

⎫ 

⎨ I 

⎬ 

̂R 2 = ⎝A 

I 

⎩ 

q 

⎠,A∈ Â, B ∈ ̂B 

B A ∗ ⎭ . 

I 

We consider the inner integration to integral (17) given by the integral 

⎛⎛ 

⎞ ⎛ ⎞ ⎞ 

∫ Z Y X I 

θ ɛ 

⎝⎝ 

I q Y ∗ ⎠ ⎝A 

I q 

⎠ x⎠ ψ V (GL2m )(Z) dZ dY dXdAdB. (18) 

Z ∗ B A ∗ I 

We consider the Fourier expansion of (18) along the abelian unipotent group that 

consists of matrices in Sp 2m(2n+1) (A) of the form 

k 2 (r) = k 2 (r 1 ,...,r 3n−1 ) = I 2m(2n+1) + 

( 3n−2 

∑ ) 

r i e ′ 2,2m+i 

+ r 3n−1 e ′ 2,4mn+1 , 

where each r j is integrated over F \A. Thus, (18) is equal to 

⎛ ⎛ 

⎞⎛ 

⎞ ⎞ 

∑ 

∫ 

Z Y X I 

θ ɛ 

⎝k 2 (r) ⎝ I q Y ∗ ⎠⎝A 

I q 

⎠ x⎠ 

α j ∈F 

Z ∗ B A ∗ I 

i=1 

× ψ V (GL2m )(Z)ψ(α 1 r 1 +···+α 3n−1 r 3n−1 ) d(···). (19) 

Let l 3 (α) = l 3 (α 1 ,...,α 3n−1 ) = I 2m(2n+1) − ∑ 3n−2 

i=1 α ie ′ 2m+i,3 − α 3n−1e ′ 4mn+1,3 .Then 

l 3 (α) ∈ ̂R 2 (F ). Using the left-invariance properties of θ ɛ , we conjugate by l 3 (α) from


left to right. Changing variables and collapsing summation with integration, integral 

(19) is equal to 

⎛⎛ 

⎞ ⎛ ⎞ ⎞ 

∫ Z Y 1 X 1 I 

θ ɛ 

⎝⎝ 

I q Y1 

∗ ⎠ ⎝A 

I q 

⎠ x⎠ ψ V (GL2m )(Z) d(···). (20) 

Z ∗ B A ∗ I 

Here Y 1 is integrated over the group Ŷ 1 generated by Ŷ and the group 

k 2 (r 1 ,...,r 3n−2 , 0), where r i ∈ A and, similarly, X 1 is in the group ̂X 1 generated 

by ̂X and the group of matrices k 2 (0,...,0,r 3n−1 ), where r 3n−2 ∈ A. Both variables 

are integrated in their groups with points in A modulo points in F . The variables A 

and B are not changed, but now we integrate the group l 3 (r 1 ,...,r 3n−1 ) with points 

in A, and all other variables in ̂R 2 are integrated with points in A modulo points in F . 

Next, we define the following group of unipotent matrices: 

{ 

( 5n−3 

∑ ) 

k 3 (r) = k 3 (r 1 ,...,r 5n−1 ) = I 2m(2n+1) + r i e ′ 3,2m+i 

i=1 

+ r 5n−2 e ′ 3,4mn+1 + r 5n−1e ′ 3,4mn+2 

Consider the Fourier expansion of (20) along the unipotent group {k 3 (r)} with points 

in A modulo points in F . Then, using 

5n−2 

∑ 

l 4 (α) = l 4 (α 1 ,...,α 5n−1 ) = I 2m(2n+1) − α i e ′ 2m+i,4 −α 5n−2e ′ 4mn+1,4 −α 5n−1e ′ 4mn+2,4 , 

i=1 

we obtain, after a suitable collapsing of summation and integration, the integral 

⎛⎛ 

⎞ ⎛ ⎞ ⎞ 

∫ Z Y 2 X 2 I 

θ ɛ 

⎝⎝ 

I q Y2 

∗ ⎠ ⎝A 

I q 

⎠ x⎠ ψ V (GL2m )(Z) d(···). (21) 

Z ∗ B A ∗ I 

Here Y 2 is integrated over the group Ŷ 2 generated by Ŷ 1 and the group 

{k 2 (r 1 ,...,r 5n−3 , 0, 0)}; similarly, X 2 is in the group ̂X 2 generated by ̂X and the 

group {k 2 (0,...,0,r 5n−2 ,r 5n−1 )}. As for the variables A and B, we integrate the 

groups {l 3 (r 1 ,...,r 3n−1 )} and {l 4 (r 1 ,...,r 5n−1 )} over A; all other variables in ̂R 2 are 

integrated with points in A modulo points in F . 

We continue this process by induction, showing that (18) is equal to 

⎛⎛ 

⎞ ⎛ ⎞ ⎞ 

∫ Z Y 2m−2 X 2m−2 I 

θ ɛ 

⎝⎝ 

I q Y2m−2 

∗ ⎠ ⎝A 

I q 

⎠ x⎠ ψ V (GL2m )(Z) d(···), (22) 

Z ∗ B A ∗ I 

} 

.


where now we integrate Y 2m−2 over the group of all matrices in Mat 2m×2m(2n−1) with the 

condition that Y 2m−2 (2m, i) = 0 for all 1 ≤ i ≤ 2m(2n − 1) − 1. The variable X 2m−2 

is integrated over the group Mat 0 2m×2m with the condition that X 2m−2(2m, 1) = 0, 

and the variables A and B are integrated over ̂R 2 (A). In this last step, that is, when 

we move from the (m − 1)th step to the mth step, we also need to use the smallness 

properties of the representation ɛ . Indeed, in this step, we first need to consider the 

Fourier coefficient along I 2m(2n+1) + re m+1,4mn+m−1 . The smallness property of the 

representation implies that the contribution to the expansion of all the nontrivial terms 

is zero. This follows from the fact that we obtain as an inner integration a Fourier 

coefficient of θ ɛ which corresponds to the unipotent orbit ((2m + 2)1 4mn−2 ). It thus 

follows from Theorem 1, adopted to this case, that these Fourier coefficients are zero. 

Applying Lemma 1, adopted to this case, to integral (22), it follows that integral 

(17) is equal to 

⎛⎛ 

⎞ 

⎞ 

∫ 

2m 

ϕ σ (g)θ N m,ψ ⎝⎝I ɛ (v 1 ,v 2 ) ′ n−1 u n−1 

⎠˜r 2˜w 2 (n)x β1 (1)˜w 1˜r 1 (g, h) ⎠ 

I 2m 

× ψ n−1 

( 

(v1 ,v 2 ) ′ n−1 u n−1) 

d(···). (23) 

Here N m is the standard unipotent radical of the parabolic subgroup of Sp 2m(2n+1) 

whose Levi part is GL 2m 

1 

× Sp 2m(2n−1) . Also, for x ∈ Sp 2m(2n+1) (A), wehave 

∫ 

θ N m,ψ 

ɛ 

(x) = θ ɛ (yx)ψ Nm (y) dy, 

N m (F )\N m (A) 

where, for y = (y i,j ) ∈ N m ,wesetψ Nm (y) = ψ(y 1,2 +···+y 2m−1,2m ). Also, in 

integral (23), the variable ˜r 2 is integrated over ̂R 2 (A), and all other variables are 

integrated as in integral (17). 

At this point, we can continue by induction on n. Indeed, notice that the Fourier 

coefficient θ N m,ψ 

ɛ 

is actually a composition of a constant term and a Whittaker coefficient. 

Indeed, the integral over N m can be computed in two steps: first, by computing 

the constant term along the unipotent radical of the parabolic subgroup of Sp 2m(2n+1) 

whose Levi part is GL 2m × Sp 2m(2n−1) ; then, by computing the Whittaker coefficient 

along the group GL 2m . From the definition of the representation ɛ , it follows that 

the above constant term, viewed as a representation of GL 2m (A) × Sp 2m(2n−1) (A), 

defines a cuspidal representation on the GL 2m -part, and on Sp 2m(2n−1) we obtain 

a representation defined on Sp 2m(2n−1) (A) which has properties similar to those of 

the residue representation on Sp 2m(2n+1) . Thus, on the group Sp 2m(2n−1) , we obtain a 

representation that corresponds to the unipotent orbit ((2m) 2n−1 ). In other words, we 

obtain a representation that has no nonzero Fourier coefficients corresponding to any


unipotent orbit greater than or not related to ((2m) 2n−1 ). Hence, in (23), we can now 

repeat the same process that we did above, but this time over the integration over 

(v 1 ,v 2 ) ′ n−1 u n−1. 

Hence, repeating this process n − 1 times, we obtain the fact that integral (23)is 

equal to 

∫ 

ϕ σ (g)θ N,ψ 

ɛ 

(˜rn+1˜w 2 (1)x βn (1)˜r n˜w 2 (2)x βn−1 (1) ···˜r 2˜w 2 (n)x β1 (1)˜w 1˜r 1 (g, h) ) d(···). 

(24) 

Here ˜r i is integrated over ̂R i , which is defined similarly to the definition of 

̂R 2 . The Weyl elements ˜w 2 (i) are defined similarly to ˜w 2 (n), and we define 

x βi (1) = diag(I 2m(i−1) ,x 

β ′ i 

(1),I 2m(i−1) ). Here, for 1 ≤ i ≤ n − 1, wesetx 

β ′ i 

(1) = 

I 2m(2n−2i+3) − e 2,n−i+3 ′ and x′ β n 

(1) = I 6m − ∑ m 

i=1 e′ 3i−1,3i . Also, the variable g is 

integrated over V (Sp 2n )(F )\Sp 2n (A). Finally, the group N is the standard maximal 

unipotent subgroup of Sp 2m(2n+1) ,and 

∫ 

θ N,ψ 

ɛ 

(x) = θ ɛ (yx)ψ N (y) dy, 

N(F )\N(A) 

where ψ N is defined as follows. For y = (y i,j ) ∈ N,define 

( 2m−1 

ψ N (y) = ψ 

∑ 

i=1 

2m−1 

∑ 

y i,i+1 + 

i=1 

2m−1 

∑ 

y 2m+i,2m+i+1 +···+ 

i=1 

y 2m(n−1)+i,2m(n−1)+i+1 

+ y 2mn+1,2mn+2 +···+y m(2n+1)−1,m(2n+1) + ay m(2n+1),m(2n+1)+1 

). 

Next, in integral (24), we factor the group V (Sp 2n ) from the g integration, obtaining 

the following basic identity. 

THEOREM 3 

Let W σ denote the Whittaker coefficient of ϕ σ . With the above notation, the integral 

∫ 

f (vh)ψ V (Sp2(m+n) ),a(v) dv 

V (Sp 2(m+n) )(F )\V (Sp 2(m+n) )(A) 

is equal to the integral 

∫ 

W σ (g)θ N,ψ 

ɛ 

(˜rn+1˜w 2 (1)x βn (1)˜r n˜w 2 (2)x βn−1 (1) ···˜r 2˜w 2 (n)x β1 (1)˜w 1˜r 1 (g, h) ) d(···), 

where g is integrated over V (Sp 2n )(A)\Sp 2n (A), and all other variables are integrated 

as in integral (24).


From this, we deduce the following. 

THEOREM 4 

Let σ denote a generic irreducible cuspidal representation of the group Sp 2n (A).Then 

the lift to Sp 2(m+n) (A) is generic. In particular, the lift is nonzero. 

Proof 

The proof is quite standard. A similar example for such a process is given, for example, 

in [GS, Section 7]. The idea is to use the identity established in Theorem 3. More 

precisely, suppose that the integral 

∫ 

f (vh)ψ V (Sp2(m+n) ),a(v) dv 

V (Sp 2(m+n) )(F )\V (Sp 2(m+n) )(A) 

is zero for every choice of data. The idea is to prove that this implies that W ϕσ (e) θɛ 

N,ψ (e) 

is zero for every choice of data. However, by our assumption that σ is generic, this 

produces a contradiction. 

From both our vanishing assumption and from Theorem 3, it follows that the 

integral 

∫ 


ɛ 

(˜rn+1˜w 2 (1)x βn (1)˜r n˜w 2 (2)x βn−1 (1) ···˜r 2˜w 2 (n)x β1 (1)˜w 1˜r 1 (g, h) ) d(···) 

is zero for every choice of data. We may assume that h = e. The idea is to follow the 

same steps that we performed to derive the identity in the statement of Theorem 3. 

So, let φ denote a Schwartz function defined on the unipotent group Ŝ(A). Since the 

above integral is zero for every choice of data, we thus obtain that for every choice of 

data, the integral 

∫ 


ɛ 

(˜rn+1˜w 2 (1)x βn (1)˜r n˜w 2 (2)x βn−1 (1) 

×···×˜r 2˜w 2 (n)x β1 (1)˜w 1˜r 1 (g, e)S ) φ(S) d(···) 

is zero. By conjugating the S variable inside the function θɛ 

N,ψ 

by changing variables, we obtain that the integral 

∫ 


ɛ 

(˜rn+1˜w 2 (1)x βn (1)˜r n˜w 2 (2)x βn−1 (1) 

×···×˜r 2˜w 2 (n)x β1 (1)˜w 1˜r 1 (g, e) )̂φ(˜r 1 ) d(···) 

from left to right and 

is zero for every choice of data. Here ̂φ is the Fourier transform of φ. Sinceφ is an 

arbitrary Schwartz function, it follows that ̂φ is also arbitrary. Thus, we deduce that


for every choice of data, the integral 

∫ 


ɛ 

(˜rn+1˜w 2 (1)x βn (1)˜r n˜w 2 (2)x βn−1 (1) ···˜r 2˜w 2 (n)x β1 (1)˜w 1 (g, e) ) d(···) 

is zero. The process is clearly inductive, and we omit the details. 

 

5. The unramified computations 

In this section, we prove that the generic part of the lift introduced in Section 4 

is functorial. From Section 3, it follows that the lift, under the assumption stated 

in Theorem 2, is cuspidal. Therefore, we can write the lift as a sum of cuspidal 

representations. From the computations of Section 4, it follows that at least one 

irreducible representation of this summand is generic. In this section, we prove that 

this summand is a functorial lift, as predicted by Definition 2. 

We use the identity established in Theorem 3; we concentrate on this case. The 

other cases are done in a similar way. Our method of proving the correspondence is the 

same in [GRS4]. At the end of this section, we suggest a different approach sketched 

out in [G2, Section 6]. This other approach has the advantage of not requiring the 

existence of a Whittaker coefficient, and it has the potential to work in other cases 

as well. In fact, using this method implies that any irreducible summand of the lift is 

functorial. 

In this section, F denotes a local nonarchimedean field. Given a group G, we 

denote by G(F ) the F -points of G; when there is no confusion, we omit F from the 

notation. 

Let π and σ denote a generic irreducible representation of Sp 2(m+n) and Sp 2n , 

respectively. We denote by ɛ the irreducible constituent of the residue representation 

constructed in Section 2. Here τ = τ(ɛ) is a generic irreducible representation of 

GL 2m which we assume to be the local lift of a generic cuspidal representation ɛ of 

SO 2m . Assume that all representations are unramified, and let W π and W σ denote 

the unramified vector of each of these two representations. Also, let θ ɛ denote the 

unramified vector in the space of ɛ . 

Assume that the above vectors satisfy the corresponding local version of the 

identity stated in Theorem 3. By this we mean the following. Clearly, the global 

Whittaker coefficient is factorizable. Also, it follows from Proposition 2, adopted to 

this case, that at the unramified places the global Fourier coefficient given by θɛ 

N,ψ 

(here, in the meaning of Theorem 3) induces a local functional that is unique. Choose 

such a functional, and if we view the residue representation at the place F as a 

quotient of an induced representation, we may realize this functional evaluated at the 

unramified vector as a function that we denote by θɛ N,ψ (x), where x ∈ Sp 2m(2n+1) . 

This function is fixed under the standard maximal compact subgroup of Sp 2m(2n+1) ;it 

is normalized so that its value at the identity is 1. In this notation, we assume that the


following identity holds: 

∫ 

W π (h) = W σ (g)θ N,ψ 

ɛ 

(˜rn+1˜w 2 (1)x βn (1)˜r n˜w 2 (2)x βn−1 (1) 

×···×˜r 2˜w 2 (n)x β1 (1)˜w 1˜r 1 (g, h) ) d(···). (25) 

We now define the local lift that corresponds to our case. Assume that π is the 

irreducible constituent of Ind Sp 2(m+n) 

B(Sp 2(m+n) ) χδ1/2 B(Sp 2(m+n) ), that σ is the irreducible constituent 

of Ind Sp 2n 

B(Sp 2n ) µδ1/2 B(Sp 2n ),andthatτ = τ(ɛ) is the irreducible constituent of 

Ind GL 2m 

B(GL 2m ) νδ1/2 B(GL 2m ). Here, for a given group G, we denote by B(G) the standard Borel 

subgroup of G. The representations χ,µ,andν are unramified characters of the 

corresponding Borel subgroups. Thus, χ is determined by n + m unramified characters 

(χ 1 ,...,χ n+m ) of F ∗ , µ by (µ 1 ,...,µ n ),andν by (ν 1 ,...,ν m ,ν −1 

1 ,...,ν−1 m ). 

Indeed, the character ν is as described since τ(ɛ) is the local lift of ɛ. To say that π 

is the local endoscopic lift from σ and ɛ is to say that the sets (χ 1 ,...,χ n+m ) and 

(µ 1 ,...,µ n ,ν 1 ,...,ν m ) are the same. 

The main result in this section is the following. 

THEOREM 5 

Let π, σ, and ɛ be as above, and suppose that integral (25) holds for all unramified 

data. Then π is the local endoscopic lift of σ and ɛ. 

Proof 

For a complex variable s, letL(π, s) denote the standard local L-function attached 

to π. Thisisa2(n + m) + 1 degree L-function. Similarly, we denote by L(σ, s) and 

L(ɛ, s) the standard L-functions of these two representations. The first L-function is 

of degree 2n + 1, and the second is of degree 2m. To prove the theorem, we prove 

that L(π, s) = L(σ, s)L(ɛ, s). 

Let h(t) = diag(t,1,...,1,t −1 ) be a torus element in Sp 2(n+m) . It follows from 

[GRS3, Theorem 3.1] that 

∫ 

( ) 

W π h(t) |t| 2s−(n+m+1/2) L(π, 2s − 1/2) 

dt = , (26) 

ζ (4s − 1) 

F ∗ 

where ζ (s) denotes the local zeta function. Thus, to prove our result, we need to prove 

that the integral 

∫ 


ɛ 

(˜rn+1˜w 2 (1)x βn (1)˜r n˜w 2 (2)x βn−1 (1) ···˜r 2˜w 2 (n) 

× x β1 (1)˜w 1˜r 1 (g, h(t)) ) |t| 2s−(n+m+1/2) d(···)


is equal to 

L(σ, 2s − 1/2)L(ɛ, 2s − 1/2) 

. 

ζ (4s − 1) 

In the above integral, we integrate t over F ∗ . Parameterize the maximal torus of Sp 2n 

as a = diag(a 1 ,...,a n ,an 

−1,...,a−1 

1 ). Performing the Iwasawa decomposition in the 

above integral, we obtain 

∫ 

W σ (a)θ N,ψ 

ɛ 

(˜rn+1˜w 2 (1)x βn (1)˜r n˜w 2 (2)x βn−1 (1) ···˜r 2˜w 2 (n) 

Here, 

× x β1 (1)˜w 1˜r 1 l ′ (a,t) ) δ B(Sp2n )(a) −1 |t| 2s−(n+m+1/2) d(···). 

l ′ (a,t) = diag(a,...,a,t,I n+m−1 ,a,I n+m−1 ,t −1 ,a,...,a), 

where the above-defined torus a occurs overall 2m − 1 times. Next, in the above 

integral, we conjugate the torus l(a,t) from right to left. We need to keep track of the 

factors obtained from the change of variables. Since, eventually, only the variables t 

and a 1 are important, the others turn out to be units; we keep track of these two only. 

From the definition of the Weyl elements ˜w 2 (j) and the groups ̂R i , it follows that 

the t-variable commutes with all ̂R i , except when i = 1. In that case, we obtain a 

factor of |t| −2n(m−1) from the change of variables. The variable a 1 contributes a factor 

of |a 1 | −4mn(m−1) from the change of variables in the ˜r 2 -variable. Since there is also a 

factor of |a 1 | −2n from the δ B(Sp2n )(a) −1 factor, the above integral is equal to 

∫ 

( ) 


ɛ l(a,t)˜rn+1˜w 2 (1)x βn (a n )˜r n˜w 2 (2)x βn−1 (a n−1 ) ···˜r 2˜w 2 (n)x β1 (a 1 )˜w 1˜r 1 

Here, 

×|a 1 | −4mn(m−1)−2n |t| 2s−(2mn+m−n+1/2) p(a 2 ,...,a n ) d(···). 

l(a,t) = diag(t,a 1 I 2m−1 , 1,a 2 I 2m−1 ,...,1,a n I 2m−1 , 

I 2m ,a −1 

n 

I 2m−1, 1,...,a −1 

1 I 2m−1,t −1 ), 

and p(a 2 ,...,a n ) is a product of factors of the form |a i | p i 

. Notice that for all 1 ≤ 

i ≤ n, the element x βi (a i ) is in the standard maximal compact subgroup of Sp 2m(2n+1) . 

Indeed, from the properties of W σ (a), it follows that its value is nonzero if and only if 

|a i /a i+1 |≤1 and |a n |≤1. Hence we may assume that |a i |≤1 for all i. By arguing 

in a way similar to the proof of Theorem 4, we deduce that we get zero contribution to 

the integral unless all variables˜r i are in the standard maximal compact group. Hence,


using the fact that |a i |≤1, the above integral is equal to 

∫ 

( ) 


ɛ l(a,t) |a1 | −4mn(m−1)−2n |t| 2s−(2mn+m−n+1/2) p(a 2 ,...,a n ) d(···). 

From the left-invariant property of θɛ 

N,ψ , it follows that we get zero contribution unless 

|a i |≥1 for all 2 ≤ i ≤ n. This, with the fact that |a i |≤1, implies that |a i |=1 for 

all 2 ≤ i ≤ n. Thus, the above integral is equal to 

∫ 

( 


ɛ l(a1 ,t) ) |a 1 | −4mn(m−1)−2n |t| 2s−(2mn+m−n+1/2) da 1 dt, 

where a = diag(a 1 , 1,...,1,a −1 

1 ) and by l(a 1,t) we now mean that we consider all 

the matrices l(a,t) as above, with the conditions a i = 1 for all 2 ≤ i ≤ n. From the 

fact that θ ɛ is a vector in the residue representation, it follows that 

( 

θ N,ψ 

ɛ l(a1 ,t) ) ( 

= W ɛ l1 (a 1 ,t) ) |t| (4mn−2n+1)/2 |a 1 | (4mn−2n+1)(2m−1)/2 , 

where l 1 (a 1 ,t) = diag(t,a −1 

1 I 2m−1), which is a matrix inside GL 2m . This follows 

from the definition of l(a,t) and from the fact that |a i |=1 for all 2 ≤ i ≤ n. Also, 

W ɛ denotes the Whittaker function of ɛ. Sinceɛ has a trivial central character, then 

W ɛ (l 1 (a 1 ,t)) = W ɛ (l 1 (1,a −1 

1 t)). Plugging this into the above equation, and changing 

variables t ↦→ ta 1 , we obtain 

∫ 

( 

W σ (a)W ɛ l1 (1,t) ) |a 1 | 2s−n−1/2 |t| 2s−m da 1 dt. 

This integral factorizes to a product of two integrals. Using (26) and the well-known 

local Whittaker integral for the standard L-function for GL 2m (see [JPS]), the above 

integral equals 

L(σ, 2s − 1/2)L(ɛ, 2s − 1/2) 

. 

ζ (4s − 1) 

Therefore, we have the identity L(π, s) = L(σ, s)L(ɛ, s) for all values of s.Fromthe 

definition of the standard local L-function, it follows that the sets (χ 1 ,...,χ n+m ) and 

(µ 1 ,...,µ n ,ν 1 ,...,ν m ) are the same. 

Another approach to prove the unramified correspondence is the one sketched in [G2, 

Section 6]. The idea is that the global integral that defines the lifting (e.g., the integral 

(3)), once we know it to be nonzero, induces a global nonzero integral given by 

∫ ∫ ∫ 

( ) 

f π (h)ϕ σ (g)θ ɛ u(g, h) ψU (u) dudg. (27) 

H (F )\H (A) H 1 (F )\H 1 (A) U(F )\U(A)


Here f π (h) is a vector in the space of π.Letπ ′ denote an irreducible summand of π. 

Assume that v is a local finite nonarchimedean place such that at that place, all data 

is unramified. As in Proposition 2, adopted to our case, we know that if (θ ɛ ) v is the 

local unramified constituent of ɛ at the local finite place v,then(θ ɛ ) v is a quotient of 

Ind M̂Q χ. Here ̂Q is a certain parabolic subgroup of H ,andχ is an unramified character 

of ̂Q. Thus, integral (27) induces a nonzero element in the space 

Hom H ×H1 

( 

(Ind 

M̂Q χ) U,ψ U 

,π ′ ⊗ σ ) , 

where (···) U,ψU denotes the twisted Jacquet module with respect to ψ U . Arguing as 

in [G2], one can show that if the above Hom space is nonzero, then any two of the 

representations π ′ ,σ,andɛ determine the third one uniquely. 

6. Liftings and poles of tensor L-functions 

6.1. Endoscopic lifting and poles of the standard tensor L-functions 

Let π denote an irreducible generic cuspidal representation of H (A),andletɛ denote 

an irreducible generic cuspidal representation of H 2 (A). Here and in what follows, 

H,H 1 ,andH 2 are as defined in Definition 2(1) – (5). Since ɛ is generic, we can 

use the result of [CKPS] to lift it to an automorphic representation τ = τ(ɛ) on 

GL k (A), where k is determined by m. We assume that τ(ɛ) is a cuspidal representation 

of GL k (A). Denote by L S (π × ɛ, s) the standard tensor L-function for the group 

H (A) × H 2 (A). Here S denotes a finite set of places, including the archimedean 

ones, such that outside of S, all data is unramified. From the references below, we 

know that these L-functions can have at most a simple pole at s = 1. Unfortunately, 

we do not know a period condition, defined on H and H 2 only, which characterizes 

the pole of these L-functions. We do, however, have a natural candidate for such a 

period, in terms of the representations π and τ(ɛ). This follows from the fact that 

L S (π × ɛ, s) = L S (π × τ(ɛ),s) and from the fact that we do have a good Rankin- 

Selberg theory for L S (π × τ(ɛ),s). We now review the global constructions for the 

tensor product L-functions, in each of the five cases. In each case, we also introduce 

the global period integral, which is related to the pole at s = 1 of this L-function. 

As before, we assume that n ≤ m. This guarantees that the constructions that we 

had in the previous sections do indeed give us nonzero cuspidal representations when 

n


(1) For general values of m and n, this case was studied in [GRS3]. The case where 

m = n was studied in [GP]. In this case, let Ẽ τ (h, s) denote the Eisenstein series defined 

on the group ˜Sp 4m (A) associated with the induced representation Ind ˜Sp 4m (A) 

˜P m 

τδ s (A) P m 

. 

Here P m is the standard parabolic subgroup of Sp 4m whose Levi part is GL 2m .Let 

θ ψ Sp 2(n+m) 

denote the theta representation defined on the group ˜Sp 2(n+m) (A). The global 

integral is then given by (see [GRS3, page 210]) 

∫ 

∫ 

ϕ π (h)θ ψ ( 

Sp 2(n+m) 

l(u)h 

)Ẽτ (uh, s)ψ Um−n (u) dudg. 

Sp 2(n+m) (F )\Sp 2(n+m) (A) U m−n (F )\U m−n (A) 

(28) 

Here U m−n is a unipotent group (denoted as H n · V 2k,k−n−1 in [GRS3]), defined as 

follows. Using the notation of Section 2.2, we consider the unipotent orbit of Sp 2(n+m) 

defined by O = ((2m − 2n)1 2(m+n) ).Asexplainedin[G3], to this unipotent orbit we 

can associate a unipotent group, denoted by U m−n and a character ψ Um−n . These are 

the unipotent group and the character that we use in the above integral. The function 

l is the projection from the group U m−n onto the Heisenberg group of 2(n + m) + 1 

variables. Arguing as in [GRS6], this Eisenstein series can have at most a simple pole 

at s 0 = (k + 2)/(2(k + 1)). We denote by Ẽ τ (h) a vector in the residue representation 

at that point. Taking the residue at s 0 in (28), we denote by P(π, τ(ɛ)) the family of 

integrals 

∫ 


∫ 

ϕ π (h)θ ψ Sp 2(n+m) 

( 

l(u)h 

)Ẽτ (uh)ψ Um−n (u) dudg. 

(2) This case is similar to case (1). The only difference is that now, π is an 

irreducible generic cuspidal representation defined on ˜Sp 2(n+m) (A), and the Eisenstein 

series is defined on the symplectic group Sp 4m (A). The corresponding global integral 

wasdefinedin[GRS3, page 210]. Denoting by E τ (h) a vector in the corresponding 

residue representation, we denote by P(π, τ(ɛ)) the family of integrals 

∫ 

∫ 

˜ϕ π (h)θ ψ ( ) 

Sp 2(n+m) 

l(u)h Eτ (uh)ψ Um−n (u) dudh. 


(3) This case was considered in [G1] and also in [So]. The case when m = n was 

studied in [GP]. We consider the Eisenstein series E τ (h, s) defined on SO 4m+1 (A), 

which corresponds to the induced representation Ind SO 4m+1(A) 

P m (A) 

τδm s . Here P m is the 

parabolic subgroup of SO 4m+1 whose Levi part is GL 2m .LetO = ( (2(m − n) + 

1)1 2(m+n)) .In[G3], we attached to this unipotent orbit a unipotent group U m−n and a


character ψ Um−n . With this notation, the global integral is given by 

∫ ∫ 

ϕ π (h)E τ (uh, s)ψ Um−n (u) dudh. 

SO 2(m+n) (F )\SO 2(m+n) (A) U(F )\U(A) 

If Re(s) > 1/2, this Eisenstein series can have at most a simple pole at a unique point 

s 0 . Denoting the residue by E τ (h), we consider the period integral 

P ( π, τ(ɛ) ) ∫ 

∫ 

= 

ϕ π (h)E τ (uh)ψ Um−n (u) dudh. 

SO 2(m+n) (F )\SO 2(m+n) (A) U m−n (F )\U m−n (A) 

(4) This case is similar to case (3). The only difference is in the rank of the 

groups in question. Once again, we can consider a period integral that is obtained by 

considering the residue of a global Rankin-Selberg integral. As above, we denote this 

period by P(π, τ(ɛ)). 

(5) This case was also considered in [G1] andin[So]. When m = n, itwas 

studied in [GP]. We consider the Eisenstein series E τ (g, s) defined on SO 4m (A) which 

τδm s . Here P m is the parabolic 

subgroup of SO 4m whose Levi part is GL 2m .Form ≥ n + 1, letO = ( (2(m − n) − 

1)1 2(m+n)+1) .In[G3], we attached to this unipotent orbit a unipotent group U m−n and a 

character ψ Um−n .Whenm = n,wesetU m−n to be the trivial group. With this notation, 

when m ≥ n + 1, the global integral is given by 

corresponds to the induced representation Ind SO 4m(A) 

P m (A) 

∫ 

SO 2(n+m)+1 (F )\SO 2(n+m)+1 (A) U m−n (F )\U m−n (A) 

∫ 

ϕ π (h)E τ (uh, s)ψ Um−n (u) dudh. 

If Re(s) > 1/2, this Eisenstein series can have at most a simple pole at a unique point 

s 0 . Denoting the residue by E τ (g), we consider the period integral 

P ( π, τ(ɛ) ) ∫ 

∫ 

= 

ϕ π (h)E τ (uh)ψ Um−n (u) dudh. 


When m = n, wedefine 

P ( π, τ(ɛ) ) = 

∫ 

SO 2(n+m) (F )\SO 2(n+m) (A) 

ϕ π (h)E τ (h) dh. 

One of the main goals of this article is to study the following.


CONJECTURE 1 

Let π denote an irreducible generic cuspidal representation of the group H (A), and let 

ɛ denote an irreducible cuspidal representation of H 2 (A) according to Section 6.1(1) – 

(5). Assume that τ = τ(ɛ) is a cuspidal representation. Then the following are 

equivalent. 

(1) The partial tensor L-function L S (π × ɛ, s) = L S (π × τ(ɛ),s) has a simple 

pole at s = 1.HereS is a finite set of places, including the archimedean ones, 

such that outside of S, all data is unramified. 

(2) There is a choice of data such that the period integral P(π, τ(ɛ)) is not zero 

for some choice of data. 

(3) There is a generic cuspidal representation σ of H 1 (A) such that π is the weak 

endoscopic lift from σ and ɛ. 

Two parts of the conjecture are, in fact, a theorem. The implication that (1) implies 

(2) follows from the usual Rankin-Selberg theory. Indeed, it follows from the above 

references that when we unfold the global integrals, we represent the above tensor 

product L-functions. It also follows from the above references that for any finite 

place, data can be chosen so that the integral is not zero. From this, it follows that 

if the partial L-function L S (π × τ(ɛ),s) has a simple pole at s = 1, the period 

integral P(π, τ(ɛ)) is not zero for some choice of data. The implication that (3) 

implies (1) follows from the definition of the weak lift. Indeed, if we assume (3), then 

L S (π × ɛ, s) = L S (σ × ɛ, s)L S (ɛ × ɛ, s). Since all data are generic, we know from 

[CKPS] that all representations have a lift to an automorphic representation of GL. 

By the result of [Sh], we know that the tensor product L-function of two automorphic 

representations does not vanish at s = 1. From this, it follows that L S (π × ɛ, s) has a 

simple pole at s = 1. 

We note that the implication that (2) implies (1) in Conjecture 1 should, in 

principle, follow also from the Rankin-Selberg integral representations given above. 

In this part, we study the implication that (2) implies (3). We use the lifting studied 

in the previous sections to prove this implication in one case. The other four cases 

stated in Conjecture 1 are different. The main problem is that the representations ɛ 

involve a representation µ(ɛ) defined on a classical group. Therefore, another step 

is required. This step requires the study of the descent of a certain residue of an 

Eisenstein series and to prove that this descent is by itself a residue. At this point, it is 

not clear how to prove this. 

THEOREM 6 

Let H and H i be as in Definition 2(5). In other words, suppose that H = SO 2(n+m)+1 , 

that H 1 = SO 2n+1 , and that H 2 = SO 2m+1 . Then Conjecture 1 holds.


Proof 

The proof of the theorem relies on Fourier expansions and uses the fact that O M ( ɛ ) = 

((2m) 2(n+1) ). These Fourier expansions are similar to those in [GRS8, proof of 

Lemma 2.4]. 

Let π denote an irreducible generic representation of H (A). We consider the 

cases when m ≥ n + 1. The case when m = n, with the assumption that π is cuspidal, 

is similar. 

It is given that the period integral 

P ( π, τ(ɛ) ) ∫ ∫ 

= 

ϕ π (h)E τ (uh)ψ Um−n (u) dudh 

H (F )\H (A) U m−n (F )\U m−n (A) 

is nonzero for some choice of data. We need to construct a generic cuspidal representation 

σ defined on H 1 (A) such that π is the endoscopic lift from σ and ɛ. Let 

σ ′ denote the automorphic representation of H 1 (A) generated by all functions of the 

form 

∫ ∫ 

( ) 

f (g) = 

ϕ π (h)θ ɛ u(g, h) ψU (u) dudh. 

H (F )\H (A) U(F )\U(A) 

Here the function θ ɛ is a vector in the space of the representation ɛ defined 

on M(A) = SO 4m(n+1) (A). This representation was constructed at the end of 

Section 2.2(5). Similarly, the function ϕ π is a vector in the space of π. 

By arguing as in Section 3, we can prove that σ ′ is a cuspidal representation of 

H 1 (A). Below, we prove that σ ′ is generic. Assuming that, let σ denote an irreducible 

cuspidal generic summand of σ ′ . Then it follows from Section 5 that π is the weak 

endoscopic lift from σ and ɛ. 

To prove that σ ′ is generic, for this proof only let R denote the standard maximal 

unipotent subgroup of H 1 = SO 2n+1 .Letψ R denote the Whittaker character defined 

on R(F )\R(A) as follows. If r = (r i,j ),thenψ R (r) = ψ(r 1,2 + ··· + r n,n+1 ). 

Assuming that P(π, τ(ɛ)) is nonzero for some choice of data, we prove that the 

integral 

∫ ∫ ∫ 

( ) 

ϕ π (h)θ ɛ u(r, h) ψU (u)ψ R (r) dr dudh (29) 

H (F )\H (A) U(F )\U(A) R(F )\R(A) 

is nonzero for some choice of data. From this, the theorem follows. 

Suppose that (29) is zero for every choice of data. We derive a contradiction. Recall 

that we may choose Weyl elements of H to be permutation matrices with zeros and 1’s. 

Given a Weyl element w, we denote by w[i, j] its (i, j)-entry. For a Weyl element to be 

in M, it is enough that if w[i, j] = 1,thenw[4m(n+1)−i+1, 4m(n+1)−j +1] = 1.


From this, it follows that a Weyl element in M is determined uniquely by the 1’s located 

in the first 2m(n+1) rows. Let w denote the Weyl element of M defined as follows. For 

all 1 ≤ j ≤ n and for all 1 ≤ i ≤ m,letw[(2j − 2)m + i, (i − 1)(2n + 1) + j] = 1. 

Let a = 2mn + 2n + 3m + 1. Forall1 ≤ j ≤ n and all 1 ≤ i ≤ m − j, set 

w[(2j − 1)m + i, a + (i − 1)(2n + 1) + j] = 1; andform − j + 1 ≤ i ≤ m, set 

w[(2j − 1)m + i, a + (i − 2)(2n + 1) + j + 1] = 1. For1 ≤ i ≤ m − n − 1, set 

w[2mn + i, 2n 2 + 2n + 1 + (i − 1)(2n + 1)],andforall2mn + m − n ≤ i ≤ 

2mn(n + 1), setw[i, i] = 1. 

In (29), we conjugate the argument of θ ɛ by the Weyl element w, andwe 

obtain 

⎛ ⎞⎛ 

⎞⎛ 

⎞ ⎞ 

∫ 

Z Y X A 

2mn 

ϕ π (h)θ ɛ ⎝ I 4m Y ∗ ⎠⎝BI 4m 

⎠⎝I u ′ h ⎠ w⎠ ˜ψ(Z, Y, B, u ′ ) d(···). 

Z ∗ CB ∗ A ∗ I 2mn 

(30) 

Here u ′ is integrated over U m−n (F )\U m−n (A), andh is integrated over H (F )\H (A). 

The character ˜ψ restricted to u ′ is equal to ψ Um−n . As matrices A, Z ∈ 

Mat 2mn×2mn ,C,X ∈ Mat 0 2mn×2mn ={L ∈ Mat 2mn×2mn : J 2mn L t =−LJ 2mn },Y ∈ 

Mat 2mn×4m ,andB ∈ Mat 4m×2mn are defined as follows. We start with the matrices 

A and Z. Write A = (A i,j ) and Z = (Z i,j ), where A i,j ,Z i,j ∈ Mat 2m×2m .First, 

we have A i,j = 0 if ij.Next,for1 ≤ i ≤ n, welet 

A i,i = I 2m ,andZ i,i is a matrix in the group of all upper triangular matrices in GL 2m . 

The precise definition of A i,j for i>jand Z i,j if j>iis not as important here as the 

relation between these two matrices. The relation is as follows. Let Z i,j (l,q) denote 

the (l,q)-entry of the matrix Z i,j .ThenifZ i,j (l,q) is a nonzero entry, A j,i (q,l) is 

zero, and vice versa. For example, if m = 2, then a possible configuration for Z i,j and 

A j,i is 

⎛ 

⎞ 

⎛ 

⎞ 

0 ∗ ∗ ∗ 

∗ ∗ ∗ ∗ 

Z i,j = ⎜0 0 0 ∗ 

⎟ 

⎝0 0 0 0⎠ , A j,i = ⎜0 ∗ ∗ ∗ 

⎟ 

⎝0 ∗ ∗ ∗⎠ , 

0 0 0 0 

0 0 ∗ ∗ 

where ∗ indicates an arbitrary nonzero entry. A similar situation holds with the pair of 

matrices B and Y and the pair X and C. Also, we mention that some of the entries in 

the variables A, B,orC lie across. By that, we mean that some variables are embedded 

inside M as a product of several 1-parameter unipotent subgroups which corresponds 

to root vectors. Finally, all variables in the above integral are integrated with points in 

A modulo points in F . 

At this point, we start with a sequence of Fourier expansions, similarly to [GRS8, 

proof of Lemma 2.4]. Using the fact that O M ( ɛ ) = ((2m) 2(n+1) ), in the same way as


in the above reference, we obtain the fact that integral (30) is equal to 

⎛ 

⎛ 

⎞ 

I 

⎛ 

⎞ 2m 

∫ 

Z 1 Y 1 X 1 

Z 2 Y 2 X 2 

ϕ π (h)θ ɛ ⎝ 

⎜ I 4mn Y1 

∗ ⎠ 

⎜ I 4m Y ∗ ⎝ 

Z1 

∗ 2 ⎟ 

⎝ 

Z2 

∗ ⎠ 

I 2m 

⎛ 

⎞ 

I 2m 

⎛ 

⎞ 

A 2 A 1 

× 

⎜ B 2 I 4m 

⎝ 

⎟ B 1 I 4mn 

⎠ 

⎝ C 2 B2 ∗ A ∗ ⎠ 

2 C 1 B1 ∗ A ∗ 1 

I 2m 

⎞ 

⎛ 

⎞ 

2mn × ⎝I 

u ′ h ⎠ 

⎟ 

˜ψ 1 (Z 1 ,Y 2 ,B 2 ,u ′ ) d(···), 

I 2mn 

⎠ 

where the variables X 2 ,Y 2 ,Z 2 and A 2 ,B 2 ,andC 2 are matrix variables inside Sp 4mn , 

defined similarly to the matrix variables X, Y, Z and A, B, C. Also, the character 

˜ψ 1 on these variables is the restriction of ˜ψ. These variables are integrated with 

points in A modulo points in F . The variable Y 1 is integrated over Mat 2m×4mn and 

Z 2 over Mat 0 2m×2m . The variable Z 1 is integrated over all upper triangular matrices 

of size 2m. All these three variables are integrated with points in A modulo points 

in F . The character ˜ψ 1 restricted to Z 1 is the Whittaker character. In other words, if 

Z 1 = (Z 1 (i, j)), then˜ψ 1 (Z 1 ) = ψ(Z 1 (1, 2) +···+Z 1 (2m − 1, 2m)). Finally, the 

variables A 1 ,B 1 ,andC 1 are integrated with points in A. 

Continuing this process inductively, this time with the corresponding matrices 

inside Sp 4mn , the above integral is equal to 

⎛ ⎛ 

⎞ ⎛ 

⎞ ⎞ 

∫ 

A 

2mn 

ϕ π (h)θ ɛ 

⎝v ⎝B 

I 4m 

⎠ ⎝I 

u ′ h ⎠ w⎠ ˜ψ 2 (v, B, u ′ ) d(···). 

C B ∗ A ∗ I 2mn 

(31) 

Here the variables A, B,andC are integrated with points in A and also v ∈ V , where 

V is the standard unipotent radical of the standard parabolic subgroup of Sp 4m(n+1) , 

whose Levi part is GL 2mn 

1 

× Sp 4m . The variable v is integrated over V (F )\V (A).The 

character ˜ψ 2 (v) is defined as follows. Write v = (v i,j ).Then˜ψ 2 (v) is defined as 

ψ(v 1,2 +···+v 2m−1,2m + v 2m+1,2m+2 +···+v 4m−1,4m 

+···+ v 2m(n−1)+1,2m(n−1)+2 +···+v 2mn−1,2mn ).


To summarize, we conclude that σ ′ is not zero if and only if integral (31) is 

nonzero for some choice of data. Arguing as in [GS] or[GJ], we first deduce that 

integral (31) is nonzero for some choice of data if and only if the integral 

⎛ ⎛ 

⎞ ⎞ 

∫ 

2mn 

ϕ π (h)θ ɛ 

⎝v ⎝vI 

u ′ h w⎠ w⎠ ˜ψ 2 (v, u ′ ) d(···) (32) 

I 2mn 

is nonzero for some choice of data. But from the definition of θ ɛ , arguing as in [GRS6], 

integral (32) is nonzero for some choice of data if and only if the integral 

∫ 

∫ 

ϕ π (h)E τ(ɛ) (uh)ψ Um−n (u) dudh 


is not zero for some choice of data. However, as stated at the beginning of the proof, 

this integral is exactly the definition of P(π, τ(ɛ)) in this case. From this, the theorem 

follows. 

 

Remark. Our construction can be extended inductively to the cases when π is a 

lift from k-distinct cuspidal representations of the classical groups corresponding to 

suitable L-group homomorphisms. It is also interesting to mention that the number of 

representations that occur are related to the poles of a certain L-function, the details 

of which we give in the case of the odd orthogonal group. 

For 1 ≤ i ≤ k, letɛ i denote a generic cuspidal representation of SO 2ni +1(A). 

Assume that n 1 +···+n k = n, and assume that all ɛ i are distinct. Then the Langlands 

conjectures predict that there exists a cuspidal generic representation π of SO 2n+1 (A), 

which is a lift from the k representations ɛ i . Clearly, our method produces these liftings 

inductively. We refer to the number k as the endoscopic number of π. 

Let ϖ 2 denote the second fundamental representation of Sp 2n (C), which is the 

L-group of SO 2n+1 (A). With the above assumptions, we have the identity 

ζ S (s)L S (π, ϖ 2 ,s) = L S (τ 1 ⊗···⊗τ k , 

2∧ ) k∏ 

,s = L 

(τ S i , 

i=1 

2∧ ) ∏ 

,s L S (τ i ⊗τ j ,s). 

i


THEOREM 7 

The irreducible cuspidal generic representation π of SO 2n+1 (A) has an endoscopic 

number k if and only if the partial L-function L S (π, ϖ 2 ,s) has a pole of order k − 1 

at s = 1. 

We mention that the endoscopic number does not determine the groups from which 

the cuspidal representation π is lifted. For example, if π is defined on SO 9 (A) and 

has an endoscopic number 1, it can be a lift either from SO 5 (A) × SO 5 (A) or from 

SO 7 (A) × SO 3 (A). 

6.2. Liftings and poles of tensor Spin L-functions 

In Section 6.1, we related the poles of standard tensor L-functions to the endoscopic 

liftings, as described in Definition 2. In this section, we relate poles of other L- 

functions to liftings and period integrals related to our global construction. Since we 

do not have a good theory of L-functions related to Spin representations, this part is 

somewhat more speculative than the previous one. 

To motivate the general conjecture, we start by considering some low-rank examples. 

Consider the following special case given in Definition 2(4). Let π denote an 

irreducible generic cuspidal representation of SO 2m+8 (A).Letɛ denote an irreducible 

generic cuspidal representation of Sp 2m (A). In Conjecture 1, we stated a criterion 

where there exists an irreducible generic cuspidal representation σ of Sp 6 (A) such 

that π is the endoscopic lift from σ and ɛ. Suppose further that σ is a lift from a 

generic cuspidal representation ν of the exceptional group G 2 (A). In this section, we 

state a conjecture analogous to this situation. 

More precisely, let π denote an irreducible generic cuspidal representation of 

GSO 2m+8 (A), andletɛ denote an irreducible generic cuspidal representation of 

GSp 2m (A). The question we study is: when can we find an irreducible generic cuspidal 

representation ν of G 2 (A) such that π is a lift from ν and ɛ corresponding to 

the homomorphism of the L-groups G 2 (C) × GSpin 2m+1 (C) ↦→ GSpin 2m+8 (C)?Itis 

convenient to summarize this by the following diagram: 

GSO 2m+8 (A) 

↑ 

GSp 6 (A) × GSp 2m (A) 

↑ 

G 2 (A) × GSp 2m (A) 

GSO 2m+8 (C) 

↑ 

GSpin 7 (C) × GSpin 2m+1 (C) 

↑ 

G 2 (C) × GSpin 2m+1 (C) 

The left-hand side of this diagram describes the lifting on the group level, and the 

right-hand side of the diagram describes the homomorphism of L-groups which corresponds 

to that lifting. In other words, let π denote an irreducible generic cuspidal 

representation π of GSO 2m+8 (A). In Section 6.1, we stated a conjecture when π is


an endoscopic lift from cuspidal generic representations of GSp 6 (A) and GSp 2m (A). 

Now, we pose another conjecture: when is π actually a lift from G 2 (A) × GSp 2m (A)? 

That is, we want to find cuspidal representations of G 2 (A) and GSp 2m (A) so that π is 

a lift from these two representations. This lift is the one associated with the L-group 

homomorphism given in the right-hand side of the above diagram. 

We have an answer to this question in the cases when m = 0 and m = 1. These 

two cases were considered in [GH1] and[GH2]. 

We start with the case when m = 0, which was studied in [GH1]. We first state 

the result and then explain the notation. 

THEOREM 8([GH1, Theorem 4.3]) 

Let π denote an irreducible generic cuspidal representation of GSO 8 (A). The following 

statements are equivalent. 

(1) The partial L-functions L S (π, St, s) and L S (π, Spin 8 ,s) both have a simple 

pole at s = 1. 

(2) The period integral Q(π, ɛ) is not zero for some choice of data. 

(3) There exists a cuspidal generic representation ν of G 2 (A) such that π is the 

weak functorial lift from ν. 

In this case, the representation ɛ is the identity representation. The L-functions considered 

in the first part are the Standard and one of the Spin representations of the 

group GSpin 8 (C). Both are of degree 8. The period Q(π, ɛ) is defined as follows. Let 

V 2 denote the standard unipotent radical subgroup of the standard maximal parabolic 

subgroup of GSO 8 whose Levi part is GL 2 × GSO 4 .LetH 9 denote the Heisenberg 

group with nine variables. Then V 2 is isomorphic to H 9 . We denote this isomorphism 

by l. Let ψ Sp 8 

denote the theta representation of ˜Sp 8 (A). WedefineQ(π, ɛ) to be the 

integral 

∫ 

∫ 

ϕ π (vk)θ ψ ( ) 

Sp 8 

l(v)k dv dk, 

SL 2 (F )×SO 4 (F )\SL 2 (A)×SO 4 (A) V 2 (F )\V 2 (A) 

where ϕ π is a vector in the space of π and θ ψ Sp 8 

is a vector in the space of ψ Sp 8 

. 

Next, we consider the case where m = 1, which was studied in [GH2]. As above, 

we first state the result. We have the following. 

THEOREM 9([GH2, Main Theorem]) 


GSO 10 (A) and GL 2 (A). The following statements are equivalent. 

(1) The partial L-function L S (π ×ɛ, Spin 10 ×Spin 3 ,s) has a simple pole at s = 1. 

(2) The period integral Q(π, ɛ) is not zero for some choice of data. 


weak functorial lift from ν and from ɛ.


In the above, the L-function is the tensor product L-function of the two Spin representations. 

Its degree is 32. In this case, ɛ is a cuspidal representation defined 

on GL 2 (A). Hence the L-group is GL 2 (C), andtheSpin 3 -representation is just the 

standard representation of that group. In this case, the period integral Q(π, ɛ) is given 

by 

∫ 

ϕ π (k)θ SO10 (k)θ ɛ (k) dk, 

SO 10 (F )\SO 10 (A) 

where θ ɛ is a vector in the space of the representation ɛ constructed in Section 2.2(4), 

with m = n = 1. 

Motivated by Theorems 8 and 9, we state the conjecture for general values of m. 

CONJECTURE 2 


GSO 2(m+4) (A) and GSp 2m (A). The following statements are equivalent. 

(1) The partial L-function L S (π × ɛ, Spin 2(m+4) × Spin 2m+1 ,s) has a simple pole 

at s = 1. 

(2) The period integral Q(π, ɛ), defined below, is not zero for some choice of data. 


weak functorial lift from ν and from ɛ. 

Here the L-function is the tensor Spin L-function whose degree is 2 2m+3 .Sincewe 

have no theory for these L-functions (except the case when m = 1), it is hard to say 

much about the relations between the first part and the others. However, we can present 

our reasoning for why we expect that (3) implies (1) in Conjecture 2. To do that, let 

η m+4 denote the (m + 4)th fundamental representation of the group Spin 2(m+4) (C). 

This is the Spin representation of that group. For 1 ≤ i ≤ m, letϖ i denote the ith 

fundamental representation of Spin 2m+1 (C), andfor1 ≤ j ≤ 3, letµ j denote the 

jth fundamental representation of Spin 7 (C). Given two representation ω 1 and ω 2 of 

the complex groups K 1 (C) and K 2 (C), respectively, we denote by (ω 1 |ω 2 ) K1 ×K 2 

the 

corresponding representation of K 1 (C) × K 2 (C). We omit the reference to K 1 and K 2 

since the group to which we are referring is clear from the context. 

To motivate the relation with the L-function in Conjecture 2, we show that 

when we restrict the representation η m+4 to the group G 2 (C) × Spin 2m+1 (C), 

then the representation (η m+4 |ϖ m ) Spin2(m+4) ×Spin 2m+1 

↓ G2 ×Spin 2m+1 

contains the identity 

representation. Indeed, it follows from [K] that branching down, we obtain the fact 

that η m+4 ↓ Spin7 ×Spin 2m+1 

= (µ 3 |ϖ m ) Spin7 ×Spin 2m+1 

. We have the identity 

ϖ m ⊗ ϖ m = 1 ⊕ 2ϖ m ⊕ m−1 

i=1 ϖ i,


from which it follows that 

(η m+4 |ϖ m ) ↓ Spin7 ×Spin 2m+1 

= (µ 3 |0) ⊕ (µ 3 |2ϖ m ) ⊕ m−1 

i=1 (µ 3|ϖ i ). (33) 

The representations on the right-hand side of this equality are representations of the 

group Spin 7 (C)×Spin 2m+1 (C).Let(01) denote the second fundamental representation 

of G 2 (C). Its degree is 7. Then µ 3 restricted to G 2 (C) gives us 1 ⊕ (01). From this, 

we obtain the fact that 

(η m+4 |ϖ m ) ↓ G2 ×Spin 2m+1 

= (0|0) ⊕ (01|0) ⊕ (0|2ϖ m ) ⊕ (01|2ϖ m ) 

⊕ m−1 

i=1 [(0|ϖ i) ⊕ (01|ϖ i )], 

where the representations of the right-hand side are representations of the group 

G 2 (C) × Spin 2m+1 (C). 

Assume that Conjecture 2(3) holds. Then the above branching decomposition 

induces a factorization of the partial L-function L S (π × ɛ, Spin 2(m+4) × Spin 2m+1 ,s), 

which contains at least one zeta factor. One expects that for generic representations, 

the other partial L-function must be nonzero at s = 1. Moreover, if the representations 

ν and ɛ are in general position, one would expect that these L-functions also must be 

holomorphic at s = 1. This would then imply that at s = 1, the pole of the above 

L-function would actually be a simple pole. This then motivates the implication that 

(3) implies (1) in Conjecture 2. 

Next, we introduce the period integral Q(π, ɛ). To simplify things, we introduce 

the period not in the similitude groups but on the orthogonal and symplectic groups 

themselves. The adoption to the similitude groups is quite simple but requires some 

more notation. Let P denote the standard parabolic subgroup of M ′ = SO 2(3m+2) 

whose Levi part is GL m−1 

2 × SO 2(m+4) ,andletU ′ denote its unipotent radical. In terms 

of matrices, this unipotent group is described in (2) with r = m − 2,p = 1, and 

q = m + 4. As explained in Section 2.2(c), one can define a projection l from the 

group U ′ onto the Heisenberg group H 4(m+4)+1 .LetK = SO 2(m+4) × SL 2 .Wedefine 

Q(π, ɛ) to be the period integral 

∫ ∫ 

ϕ π (k 1 )θ ψ ( 

Sp l(u)(k1 

4(m+4) 

,k 2 ) ) θ ɛ( ′ u(k1 ,k 2 ) ) ψ U ′(u) dudk 1 dk 2 . (34) 

K(F )\K(A) U ′ (F )\U ′ (A) 

Here k 1 ∈ SO 2(m+4) ,andk 2 ∈ SL 2 . The function θ 

ɛ ′ is a vector in the space of the 

representation ′ ɛ defined on the group M′ (A) as follows. Recall the representation 

ɛ defined on the group M = SO 14m+8 in Section 2.2(4), with n = 3. This representation 

is a residue representation that was attached to the induced representation 

from the parabolic subgroup Q whose Levi part is GL 3 2m+1 × SO 2(m+1).Todefinethe 

representation ′ ɛ , we start with the parabolic subgroup Q′ of M ′ whose Levi part


is GL 2m+1 × SO 2(m+1) . Then we attach to the corresponding induced representation 

an Eisenstein series whose residue we denote by ′ ɛ 

. All other notation is as in 

Section 2.3(c). Arguing as in [GRS1, pages 114 – 115], by choosing suitable Schwartz 

functions in the representation θ ψ Sp 4(m+4) 

, the integral (34) converges absolutely. We now 

prove the following theorem. 

THEOREM 10 

Conjecture 2(2) implies Conjecture 2(3). 

Proof 

The case when m = 1 was shown in [GH2] in complete detail. For general values 

of m, the proof is similar and follows similar steps as in the proof of Theorem 6. 

Therefore, we sketch only the main steps. 

The key ingredient is to use the construction of the lifting from the group Sp 6 × 

Sp 2m to the group SO 2(m+4) . This lifting was introduced in Section 2.3(c), where 

we now take n = 3. LetM = SO 14m+8 ,andletH = SO 2(m+4) . Starting with the 

representations π and ɛ as above, we construct an automorphic representation σ ′ , 

defined on the group Sp 6 (A), as the space generated by all functions 

∫ ∫ 

f (g) = 

ϕ π (h)θ ψ ( ) ( ) 

Sp 12(m+4) 

l(u)(g, h) θτ(ɛ) u(g, h) ψU (u) dudh. 

H (F )\H (A) U(F )\U(A) 

As in Section 3, we can prove that σ ′ is a cuspidal representation of the group Sp 6 (A). 

In fact, the computations are very similar to those in [GH2, Lemma 8], where the case 

m = 1 was studied in detail. If nonzero, then it follows from Section 5 that if σ is 

any nonzero irreducible summand of σ ′ , π is an endoscopic lift from σ and ɛ. Hence 

we need to know when σ ′ is nonzero and when it is a lift from the exceptional group 

G 2 (A). For that, we use the result from [GJ], described as follows. Let σ denote an 

irreducible cuspidal representation of Sp 6 (A). Suppose that for some vector ϕ σ in the 

space of σ , the period integral 

⎛⎛ 

⎞ ⎛ ⎞⎞ 

∫ ∫ I 1 X Y k 1 

ϕ ψ σ (g) = 

ϕ σ 

⎝⎝ 

I 2 X ∗ ⎠ ⎝ k 1 

⎠⎠ ψ(tr X) dXdY dk 1 

SL 2 (F )\SL 2 (A) (F \A) I 7 2 k 1 

is nonzero. Here X ∈ Mat 2×2 ,andY ∈ Mat 0 2×2 

. Then it follows from [GJ] that there 

exists a cuspidal generic representation ν of the exceptional group G 2 (A) such that σ 

is the weak functorial lift from ν. 

Performing steps similar to those in the proof of Theorem 6, we deduce that the 

above integral is nonzero if and only if Q(π, ɛ) is nonzero.


To motivate a possible generalization of Conjecture 2, we first extend the branching rule 

(33). Indeed, let η m+l denote the (m+l)th fundamental representation of Spin 2(m+l) (C), 

and for 1 ≤ i ≤ m,letϖ i denote the ith fundamental representation of Spin 2m+1 (C). 

For 1 ≤ j ≤ l − 1, letµ j denote the jth fundamental representation of Spin 2l−1 (C). 

Generalizing the branching rule (33) by using, for example, the method of [K], we 

obtain 

(η m+l |ϖ m ) ↓ Spin2l−1 ×Spin 2m+1 

= (µ l−1 |0) ⊕ (µ l−1 |2ϖ m ) ⊕ m−1 

i=1 (µ l−1|ϖ i ). (35) 

Suppose that Spin 2l−1 has a subgroup such that when restricting the Spin representation 

to this group, we get a fixed vector. Then the representation (η m+l |ϖ m ) ↓ Spin2l−1 ×Spin 2m+1 

has a fixed vector. Motivated by that, we state the following. 

CONJECTURE 3 


GSO 2(m+l) (A) and GSp 2m (A). The following statements are equivalent. 

(1) The partial L-function L S (π × ɛ, Spin 2(m+l) × Spin 2m+1 ,s) has a simple pole 

at s = 1. 

(2) The period integral Q ′ (π, ɛ) is not zero for some choice of data. 

(3) There exists a cuspidal generic representation ν of GSp 2(l−1) (A) such that π is 

the weak functorial lift from ν and from ɛ. 

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School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel-Aviv University, 

Ramat-Aviv 69978, Israel; ginzburg@post.tau.ac.il

A CHARACTERIZATION OF SUBSPACES AND 

QUOTIENTS OF REFLEXIVE BANACH SPACES 

WITH UNCONDITIONAL BASES 

W. B. JOHNSON and BENTUO ZHENG 

Abstract 

We prove that the dual or any quotient of a separable reflexive Banach space with the 

unconditional tree property (UTP) has the UTP. This is used to prove that a separable 

reflexive Banach space with the UTP embeds into a reflexive Banach space with an 

unconditional basis. This solves several longstanding open problems. In particular, 

it yields that a quotient of a reflexive Banach space with an unconditional finitedimensional 

decomposition (UFDD) embeds into a reflexive Banach space with an 

unconditional basis. 


It has long been known that Banach spaces with unconditional bases as well as their 

subspaces are much better behaved than general Banach spaces and that many of the 

reflexive spaces (including L p (0, 1), 1

506 JOHNSON and ZHENG 

There is, of course, quite a lot known concerning problems (a) and (b). For 

example, Pełczyński and Wojtaszczyk [15, Theorem 1.1] proved that if X has an 

unconditional expansion of identity (i.e., a sequence (T n ) of finite-rank operators such 

that ∑ T n converges unconditionally in the strong operator topology to the identity 

on X), then X is isomorphic to a complemented subspace of a space that has an 

unconditional finite-dimensional decomposition (UFDD). Later, Lindenstrauss and 

Tzafriri [11, Theorem 1.g.5] showed that every space with a UFDD embeds (not 

necessarily complementably) into a space with an unconditional basis. As regards 

reflexive spaces, it was shown in [4, Theorem 3.1] using a result from [1, Lemma 1] 

(and answering a question from that article), that if a reflexive Banach space embeds 

into a space with an unconditional basis, then it embeds into a reflexive space with 

an unconditional basis. As regards the quotient problem mentioned above, Feder [3, 

Theorem 4] gave a partial solution by proving that if X is a quotient of a reflexive 

space that has a UFDD and X has the approximation property, then X embeds into a 

space with an unconditional basis. 

It is well known and easy to see that if a Banach space X embeds into a space 

with an unconditional basis, then X has the unconditional subsequence property; 

that is, there exists a K > 0 such that every normalized weakly null sequence in 

X has a subsequence that is K-unconditional. In fact, failure of the unconditional 

subsequence property is the only known criterion for proving that a given reflexive 

space does not embed into a space with an unconditional basis. However, in Section 

3, we construct a Banach space that has the unconditional subsequence property 

but does not embed into a Banach space that has an unconditional basis. This is not 

surprising, given previous examples of Odell and Schlumprecht [12]. Moreover, Odell 

and Schlumprecht have taught us that when a subsequence property is replaced with 

the corresponding “branch of a tree” property (see [12, introduction]), the result is a 

stronger property that sometimes can be used to give a characterization of spaces that 

embed into a space with some kind of structure. The relevant property for us here is 

the unconditional tree property (UTP), and Odell and Schlumprecht’s beautiful results 

are essential tools for us in applying it. 

We use standard Banach space theory terminology, such as can be found in [11]. 

2. Main results 


Let [N]

A CHARACTERIZATION OF SUBSPACES AND QUOTIENTS 507 

ordered subset of the tree under the tree order. We say that X has the C-UTP if every 

normalized weakly null tree in X has a C-unconditional branch for some C>0 and 

that X has the UTP if X has the C-UTP for some C>0. 

Remark 2.2 

Odell, Schlumprecht, and Zsákprovedin[14, Proposition 2.2] that if every normalized 

weakly null tree in X admits a branch that is unconditional, then X has the C-UTP 

for some C > 0. A simpler proof appears in the preprint of Haydon, Odell, and 

Schlumprecht [5]. There is, therefore, no ambiguity when using the term UTP. 

Given a finite-dimensional decomposition (FDD) (E n ), (x n ) is said to be a block 

sequence with respect to (E n ) if there exists a sequence of integers 0 = m 1


to zero. Then there are blockings (B 

n ′ ) of (B n) and (C 

n ′ ) of (C n) such that for any further 

blockings ( B˜ 

n ) of (B 

n ′ ) with B˜ 

n = ⊕ k n+1 −1 

i=k n 

B 

i ′ and ( C˜ 

n ) of (C 

n ′ ) with C˜ 

n = ⊕ k n+1 −1 

i=k n 

C 

i 

′ 

and for any x ∈ B˜ 

n ,thereisay ∈ ˜C n−1 ⊕ C˜ 

n such that ‖Tx− y‖ ≤δ n ‖x‖. 

Proof 

Let (δ i ) be a sequence of positive numbers decreasing to zero. Let ( ˜δ i ) be another 

sequence of positive numbers that go to zero so fast that ∑ ∞ 

˜ 

j=i 

δ j


([12, Theorem 3.3]) there is a blocking (F n ) of the (E n ) which is a USB FDD. Then we 

use the “killing the overlap” technique of [6] to get a further blocking (G n ) so that any 

norm 1 vector y is a small perturbation of the sum of a skipped block sequence (y i ) 

with respect to (F n ) and y i ∈ G i−1 ⊕ G i .LetQ : Y ↦→ X be the quotient map. Using 

Lemma 2.5 and passing to a further blocking, without loss of generality, we assume 

that QG i is essentially contained in H i−1 + H i , where (H i ) is the corresponding 

blocking of (V n ).Let(x A ) be a normalized weakly null tree in X. We then choose a 

branch (x Ai ) so lacunary that (x Ai ) is a small perturbation of a block sequence of (H n ), 

and for each i there is at least one H ki between the essential support of x Ai and x Ai+1 . 

Let x = ∑ a i x Ai with ‖x‖ =1. Considering a preimage y of x under the quotient 

Q from Y onto X (with ‖y‖ =1), by our construction we can essentially write y as 

the sum of (y i ), where (y i ) is a skipped block sequence with respect to (F n ).Since 

(F n ) is a USB, (y i ) is unconditional. By passing to a suitable blocking (z i ) of (y i ) and 

then using Lemma 2.5, it is not hard to show that Qz i is essentially equal to a i x Ai . 

Noticing that (z i ) is unconditional, we conclude that (x Ai ) is also unconditional. 

For the general case where X and Y do not have an FDD, we have to embed them 

into some superspaces with FDD. The difficulty is that when we decompose a vector 

in Y as the sum of disjointly supported vectors in the superspace, we do not know that 

the summands are in Y . The same problem occurs for vectors in X. This makes the 

proof rather technical and requires many computations. 

THEOREM 2.8 

Let X be a quotient of a separable reflexive Banach space Y with the UTP. Then X 

has the UTP. 

Proof 

By Zippin’s result (see [17]), Y embeds isometrically into a reflexive space Z with 

an FDD. A key point in the proof is that Odell and Schlumprecht proved (see [13, 

Proposition 2.4]) that there is a further blocking (G n ) of the FDD for Z, δ = (δ i ),and 

a C>0 such that every δ-skipped block sequence (y i ) ⊂ Y with respect to (G i ) is 

C-unconditional. Let λ be the basis constant for (G n ). 

Since X is separable, we can regard X as a subspace of L ∞ .Letɛ>0. We may 

assume that 

∑ 

(a) 

j>i δ j


Let (x A ) be a normalized weakly null tree in X. Thenwelet(E n ) and (F n ) be 

blockings of (G i ) and (v i ), respectively, which satisfy the conclusions of Lemmas 2.5 

and 2.6. Using the “killing the overlap” technique (see [13, Proposition 2.6]), we can 

find a further blocking ( E˜ 

n = ⊕ l(n+1) 

i=l(n)+1 E i) 

so that for every y ∈ SY , there exist 

(y i ) ⊂ Y and integers (t i ) with l(i − 1)


This gives an estimate of the second term. For the third term, we have 

∥ 

∥a i x Ai − 

( k 2i−1−1 

∑ 

P˜ 

j 

)x∥ < ∥ 

( k 2i−1−1 

∑ 

P˜ 

∥ 

j 

)(a i x Ai − x) ∥ + ∥a i x Ai − 

( k 2i−1−1 

∑ 

∥ 

P˜ 

∥∥ 

j 

)x Ai 

j=k 2i−2 

j=k 2i−2 

< 2 

(k 2i−2 δ k2i−2 + ∑ ) 

δ j + 2δ i 

j≥k 2i−1 

j=k 2i−2 

< 2(δ k2i−2 −1 + δ k2i−1 −1) + 2δ i < 4δ i . (2.4) 

For the first term, let Q j be the canonical projection from X onto F j ,andletJ 1 = 

[t k2i−3 +1,t k2i−1 +1],J 2 = [l k2i−2 + 1,l k2i−1 ],andJ 1 ′ = (t k 2i−3 +1,t k2i−1 +1). Thenwehave 

( 

∥ 

∑ ) ( ∑ ) 

∥Q P j y − Q j Qy∥ 

j∈J 1 j∈J 2 ( 

∥ ∑ ) ( ∑ ) 

( ∑ ) ( ∑ ) 

≤ ∥Q P j y − Q j Qy∥ + ∥ Q j Qy − Q j Qy∥ 

j∈J 1 j∈J 1 j∈J 1 j∈J 2 ( 

∥ 

∑ ) ( ∑ ) 

= ∥Q P j y − Q j Qy∥ + ∥ ∑ ( ∑ )∥ ∥∥ 

Q j ai x Ai 

j∈J 1 j∈J 1 j∈J 1 −J 2 ( 

∥ 

∑ ) ( ∑ ) 

< ∥Q P j y − Q j Qy∥ + 4δ i 

j∈J 1 j∈J 1 ( ≤ 

∑ ) ( ∑ )∥ ∥( ∥∥ ∥∥ 

∑ ) ( ∑ )∥ ∥∥ 

∥ Q j Q P j y + Q j Q P j y + 4δi 

j/∈J 1 j∈J 1 j∈J 1 j/∈J 1 ( < ∥ 

∑ ( ∑ )∥ ∥( ∥∥ ∥∥ 

∑ ( ∑ )∥ ∥∥ 

Q P j y + Q j 

)Q 

+ 6δi 

j/∈J 1 

Q j 

) 

j∈J ′ 1 

j∈J ′ 1 

j/∈J 1 

P j y 

< 2λδ i + 2λδ i + 6δ i = (4λ + 6)δ i . (2.5) 

From inequalities (2.2)–(2.5), we conclude that 

‖Qz i − a i x Ai ‖ < (4λ + 12)δ i . 

Let (ɛ i ) ⊂{−1, 1} N .LetI ⊂ N be the set of indices i ∈ N for which ‖y i ‖


Remark 2.9 

If the original space Y has the (1 + ɛ)-UTP for any ɛ>0, then any quotient of Y has 

the (1 + ɛ)-UTP for any ɛ>0. 

The following is an elementary lemma, which is used later. We omit the standard 

proof. 

LEMMA 2.10 

Let X be a Banach space, and let X 1 ,X 2 be two closed subspaces of X.IfX 1 ∩X 2 ={0} 

and X 1 + X 2 is closed, then X embeds into X/X 1 ⊕ X/X 2 . 

In [7, Theorem 4.4], Johnson and Rosenthal proved that any separable Banach space 

X admits a subspace Y so that both Y and X/Y have an FDD. The proof uses 

Markuschevich bases; a Markuschevich basis for a separable Banach space X is a 

biorthogonal system {x n ,xn ∗} n∈N for which the span of the x n ’s is dense in X and the 

xn ∗ ’s separate the points of X.By[11, Theorem 1.f.4], every separable Banach space X 

has a Markuschevich basis {x n ,xn ∗} n∈N so that [xn ∗ ] contains any designated separable 

subspace of X ∗ . The following lemma is a stronger form of the result of Johnson and 

Rosenthal, which follows from the original proof. For the convenience of the reader, 

we give a sketch of the proof. We use [x i ] i∈I to denote the closed linear span of (x i ) i∈I . 

LEMMA 2.11 

Let X be a separable Banach space. Then there exists a subspace Y with FDD (E n ) such 

that for any blocking (F n ) of (E n ) and for any sequence (n k ) ⊂ N, X/span{(F nk ) ∞ k=1 } 

admits an FDD (G n ). Moreover, if X ∗ is separable, (E n ) and (G n ) can be chosen to 

be shrinking. 

Proof 

Let {x i ,xi ∗} be a Markuschevich basis for X so that [x∗ i 

] is a norm-determining 

subspace of X ∗ and even [xi ∗] = X∗ if X ∗ is separable. Then we can choose inductively 

finite sets σ 1 ⊂ σ 2 ⊂··· and η 1 ⊂ η 2 ⊂··· so that σ = ⋃ ∞ 

n=1 σ n and η = ⋃ ∞ 

n=1 η n 

are complementary infinite subsets of the positive integers and for n = 1, 2,..., 

(i) if x ∗ ∈ [xi ∗] i∈η n 

, there is an x ∈ [x i ] i∈ηn ∪σ n+1 

such that ‖x‖ =1 and |x ∗ (x)| > 

(1 − 1/(n + 1))‖x ∗ ‖; 

(ii) if x ∈ [x i ] i∈σn , there is an x ∗ ∈ [xi ∗] i∈σ n ∪η n 

such that ‖x ∗ ‖=1 and |x ∗ (x)| > 

(1 − 1/(n + 1))‖x‖. 

Once we have this, by [7, proof of Theorem 4] we have it that [x i ] ⊥ i∈σ is the w∗ - 

closure of [xi ∗] i∈η. PutY = [xi ∗]⊥ i∈η 

= [x i] i∈σ . By the analogue of [7, Proposition 

2.1(a)], we deduce that X/Y has an FDD and that ([x i ] i∈σn ) ∞ n=1 forms an FDD for 

Y . In order to prove Lemma 2.11, it is enough to prove that for any blocking ( n ) of 

(σ n ) or any subsequence (σ nk ) of (σ n ) (this, of course, needs the redefining of (η n )),


(i) and (ii) still hold. But this is more or less obvious because if n = ⋃ k n 

i=k n−1 +1 σ i, 

then we define n = ⋃ k n 

i=k n−1 +1 η i and it is easy to check that { n , n } satisfy (i) 

and (ii). For a subsequence (σ nk ),ifwelet k = σ nk and define k = ⋃ n k+1 −1 

i=n k 

η i , 

then { n , n } satisfy (i) and (ii). The rest is exactly the same as in [7, proof of Theorem 

4.4]. 

 

The next lemma shows that for a reflexive space with a USB FDD, its dual also has a 

USB FDD. 

LEMMA 2.12 

Let X be a reflexive Banach space with a USB FDD (E n ). Then there is a blocking 

(F n ) of (E n ) such that (F ∗ n ) isaUSBFDDforX∗ . 

Proof 

Without loss of generality, we assume that (E n ) is monotone. Let (δ i ) be a sequence 

of positive numbers decreasing fast to zero. By the “killing the overlap” technique, 

we get a blocking (F n ) of (E n ) with F n = ∑ k n 

i=k n−1 +1 E i so that given any x = ∑ x i 

with x i ∈ E i , ‖x‖ =1, there is an increasing sequence (t n ) with k n−1


where C is the unconditional constant associated with the USB FDD (E n ).Ifwelet 

∑ 

δi 

1 − ɛ 

C(1 + ɛ) . 

Therefore (xi ∗ ) is unconditional with unconditional constant less than (1 + 3ɛ)C if ɛ 

is sufficiently small. Hence (Fn ∗ ) is a USB FDD. 

 

THEOREM 2.13 

Let X be a separable reflexive Banach space. Then the following are equivalent. 

(a) X has the UTP. 

(b) X embeds into a reflexive Banach space with a USB FDD. 

(c) X ∗ has the UTP. 

Proof 

It is obvious that (b) implies (a). If we can prove that (a) implies (b) and that X satisfies 

(b), then by Lemma 2.12, X ∗ is a quotient of a reflexive space with a USB FDD. So, 

by Theorem 2.8, X ∗ has the UTP. Hence we need only show that (a) implies (b). Let 

X 1 be a subspace of X with an FDD (E n ) given by Lemma 2.11.By[13, Proposition 

2.4], we get a blocking (F n ) of (E n ) so that (F n ) is a USB FDD. Let Y 1 = [F 4n ],and 

let Y 2 = [F 4n+2 ].Then(F 4n ) and (F 4n+2 ) form UFDDs for Y 1 and Y 2 . By Lemma 2.11, 

X/Y i has an FDD. Since X has the UTP, by Theorem 2.8 we know that X/Y i has 

the UTP. By using [13, Proposition 2.4] again, we know that X/Y i has a USB FDD. 

Noticing that Y 1 ∩ Y 2 ={0} and that Y 1 + Y 2 is closed, by Lemma 2.10 we have that 

X embeds into X/Y 1 ⊕ X/Y 2 . Hence X embeds into a reflexive space with a USB 

FDD. 

 

COROLLARY 2.14 

Let X be a separable reflexive Banach space with the UTP. Then X embeds into a 

reflexive Banach space with an unconditional basis. 

Proof 

By Theorem 2.13, X embeds into a reflexive space Y with a USB FDD (E n ).We 

prove that Y embeds into a reflexive space with a UFDD. Then, as mentioned in the 

introduction, Y embeds into a reflexive space with an unconditional basis, and so X 

does, too. 

By Lemma 2.12, there is a blocking (F n ) of (E n ) such that (Fn ∗ ) is a USB FDD for 

Y ∗ .Now,letY 1 = ⊕ F 4n ,andletY 2 = ⊕ F 4n+2 .ThenwehaveY 1 ∩ Y 2 ={0}, and 

Y 1 + Y 2 is closed because (F 2n ), being a skipped blocking of (E n ), is unconditional. 

By Lemma 2.10, Y embeds into Y/Y 1 ⊕ Y/Y 2 .Since(Y/Y i ) ∗ is isomorphic to Yi 

⊥ ,it


is enough to prove that Yi 

⊥ has a UFDD. Let G ∗ n = F 4n−3 ∗ ⊕ F 4n−2 ∗ ⊕ F 4n−1 ∗ .Itiseasy 

to see that (G ∗ n ) forms an FDD for Y 1 ⊥. Noticing that (G n) is a skipped blocking of 

(Fn ∗), we conclude that (G n) is unconditional. Similarly, we can show that Y2 ⊥ admits 

a UFDD. This finishes the proof. 

 


Let X be a quotient of a reflexive Banach space with a UFDD. Then X embeds into a 

reflexive Banach space with an unconditional basis. 

Proof 

Combine Theorem 2.8 and Corollary 2.14. 

 

We mention again that in 1974, Davis, Figiel, Johnson, and Pełczyński proved in [1] 

that a reflexive Banach space X that embeds into a Banach space with a shrinking 

unconditional basis embeds into a reflexive space X with an unconditional basis. 

The next year, Figiel, Johnson, and Tzafriri [4] got a stronger result by removing the 

shrinkingness of the unconditional basis in the hypothesis. Our next corollary gives a 

parallel result for quotients. 


Let X be a separable reflexive Banach space. If X is a quotient of a Banach space 

with a shrinking unconditional basis, then X is isomorphic to a quotient of a reflexive 

Banach space with an unconditional basis. 

Proof 

Since X is a quotient of a Banach space with a shrinking unconditional basis, X ∗ is a 

subspace of a Banach space with an unconditional basis. Hence by [4, Theorem 3.1], 

X ∗ is isomorphic to a subspace of a reflexive Banach space with an unconditional 

basis. Therefore, X is isomorphic to a quotient of a reflexive Banach space with an 


 

Remark 2.17 

Corollary 2.16 is different from the result of [4] in that the shrinkingness in our result 

cannot be removed. The reason is more or less obvious since every separable Banach 

space is a quotient of l 1 , which has an unconditional basis. 

Gluing Theorem 2.13 and Corollaries 2.14, 2.15, and2.16 together, we have the 

following long list of equivalences. 

THEOREM 2.18 

Let X be a separable reflexive Banach space. Then the following are equivalent.


(a) 

(b) 

(c) 

(d) 

(e) 

(f) 

(g) 

(h) 

(i) 

X has the UTP. 

X is isomorphic to a subspace of a Banach space with an unconditional basis. 

X is isomorphic to a subspace of a reflexive space with an unconditional basis. 

X is isomorphic to a quotient of a Banach space with a shrinking unconditional 

basis. 

X is isomorphic to a quotient of a reflexive space with an unconditional basis. 

X is isomorphic to a subspace of a quotient of a reflexive space with an 


X is isomorphic to a subspace of a reflexive quotient of a Banach space with a 

shrinking unconditional basis. 

X is isomorphic to a quotient of a subspace of a reflexive space with an 


X is isomorphic to a quotient of a reflexive subspace of a Banach space with a 

shrinking unconditional basis. 

3. Example 

In this section, we give an example of a reflexive Banach space for which there exists 

a C>0 such that every normalized weakly null sequence admits a C-unconditional 

subsequence, while for any D>0 there is a normalized weakly null tree such that 

every branch is not D-unconditional. The construction is an analogue of Odell and 

Schlumprecht’s example (see [12, Example 4.2]). 

We first construct an infinite sequence of reflexive Banach spaces X n . Each X n is 

infinite-dimensional and has the property that for ɛ>0, every normalized weakly null 

sequence has a (1 + ɛ)-unconditional basic subsequence, while there is a normalized 

weakly null tree for which every branch is at least C n -unconditional and C n goes to 

infinity when n goes to infinity. Then the l 2 -sum of X n ’s is a reflexive Banach space 

with the desired property. 

Let [N] ≤n be the set of all subsets of the positive integers with cardinality less than 

or equal to n. Letc 00 ([N] ≤n ) be the space of sequences with finite support indexed 

by [N] ≤n , and denote its canonical basis by (e A ) A∈[N] ≤n. Let(h i ) be any normalized 

conditional basic sequence that satisfies a block lower l 2 -estimate with constant 1,for 

example, the boundedly complete basis of James’s space (see [2, Problem 6.41]). Let 

∑ 

aA e A be an element of c 00 ([N] ≤n ).Let(β k ) m k=1 be disjoint segments. By “a segment 

in [N] ≤n ,” we mean a sequence (A i ) k i=1 ∈ [N]≤n with A 1 ={n 1 ,n 2 ,...,n l },A 2 = 

{n 1 ,n 2 ,...,n l ,n l+1 },...,A k ={n 1 ,n 2 ,...,n l ,...,n l+k−1 } for some n 1


Let X = ( ∑ X n 

)2 .LetC M be the unconditional constant of (h i ) M i=1 . It is clear that C M 

tends to infinity when M goes to infinity. The normalized weakly null tree (e A ) A∈[N] ≤M 

in X M has the property that every branch of it is 1-equivalent to (h i ) M i=1 since (h i) has 

a block lower l 2 -estimate with constant 1. So what is remaining is to verify that for 

every ɛ>0, every normalized weakly null sequence in X has a (1 + ɛ)-unconditional 

basic subsequence. Actually, we prove that there is a subsequence which is (1 + ɛ)- 

equivalent to the unit vector basis of l 2 . By a gliding-hump argument, it is not hard to 

verify the following fact. 

Fact 

Let (Y k ) be a sequence of reflexive Banach spaces, and let Y = ( ∑ Y k 

)l 2 

. If for every 

ɛ>0,k ∈ N, every normalized weakly null sequence in Y k has a subsequence that is 

(1+ɛ)-equivalent to the unit vector basis of l 2 , then for every ɛ>0, every normalized 

weakly null sequence in Y has a subsequence that is (1 + ɛ)-equivalent to the unit 

vector basis of l 2 . 

Considering this fact, it is enough to show that for every ɛ > 0,k ∈ N, every 

normalized weakly null sequence in X k has a subsequence that is (1 + ɛ)-equivalent 

to the unit vector basis of l 2 . We prove this by induction. 

For k = 1, X 1 is isometric to l 2 , so the conclusion is obvious. 

Assume that the conclusion is true for X k . By the definition of X k+1 , X k+1 is 

isometric to ( ∑ (R ⊕ X k ) ) l 2 

(where R ⊕ X k has some norm so that {0} ⊕X k is 

isometric to X k ). Hence by hypothesis and the fact mentioned above, it is easy to see 

that the conclusion is true in X k+1 . This finishes the proof. 

Remark 3.1 

The proof of the corresponding induction step in [12, Example 4.2] is more complicated 

than the very simple induction argument in the previous paragraph. Schlumprecht 

realized after [12] was published that the induction could be done this simply (see 

[16]), and his argument works in our context. 

Acknowledgments. The authors thank the referees for useful corrections, especially 

for pointing out the imprecision in the initial construction of the example in Section 

3. This article is based in part on the doctoral dissertation of Zheng, which is being 

prepared at Texas A&M University under Johnson’s direction. 

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Johnson 

Department of Mathematics, Texas A&M University, College Station, Texas 77843, USA; 

johnson@math.tamu.edu 

Zheng 

Department of Mathematics, University of Texas at Austin, Austin, Texas 78712, USA; 

btzheng@math.utexas.edu

DEGREE GROWTH OF MEROMORPHIC 

SURFACE MAPS 

SÉBASTIEN BOUCKSOM, CHARLES FAVRE, and MATTIAS JONSSON 

Abstract 

We study the degree growth of iterates of meromorphic self-maps of compact Kähler 

surfaces. Using cohomology classes on the Riemann-Zariski space, we show that the 

degrees grow similarly to those of mappings that are algebraically stable on some 

bimeromorphic model. 


Let X be a compact Kähler surface, and let F : X X be a dominant meromorphic 

mapping. Fix a Kähler class ω on X, normalized by (ω 2 ) X = 1, and define the degree 

of F with respect to ω to be the positive real number 

deg ω (F ):= (F ∗ ω · ω) X = (ω · F ∗ ω) X , 

where (·) X denotes the intersection form on H 1,1 

R 

(X). WhenX = P2 and ω is the 

class of a line, this coincides with the usual algebraic degree of F . One can show that 

deg ω (F n+m ) ≤ 2deg ω (F n )deg ω (F m ) for all m, n. Hence the limit 

λ 1 := lim 

n→∞ 

deg ω (F n ) 1/n 

exists. We refer to it as the asymptotic degree of F . It follows from standard arguments 

(see Proposition 3.1) thatλ 1 does not depend on the choice of ω, thatλ 1 is invariant 

under bimeromorphic conjugacy, and that λ 2 1 ≥ λ 2, where λ 2 is the topological degree 

of F . 

MAIN THEOREM 

Assume that λ 2 1 >λ 2. Then there exists a constant b = b(ω) > 0 such that 

deg ω (F n ) = bλ n 1 + O(λn/2 2 ) as n →∞. 


Vol. 141, No. 3, c○ 2008 DOI 10.1215/00127094-2007-004 

Received 28 August 2006. Revision received 22 May 2007. 

2000 Mathematics Subject Classification. Primary 32H50; Secondary 14E05, 14C17. 

Boucksom’s work supported in part by the Japanese Society for the Promotion of Science. 

Jonsson’s work supported in part by National Science Foundation grant DMS-0449465, the Swedish Research 

Council, and the Gustafsson Foundation. 

519

520 BOUCKSOM, FAVRE, and JONSSON 

The dependence of b on ω can be made explicit (see Remark 3.7). For the polynomial 

map F (x,y) = (x d ,x d y d ) on C 2 (with ω the standard Fubini-Study form), one has 

λ 2 = λ 2 1 = d2 , deg ω (F n ) = nd n ; hence the assertion in the main theorem may fail 

when λ 2 1 = λ 2. 

Degree growth is an important component in the understanding of the complexity 

and dynamical behavior of a self-map and has been studied in a large number of works 

in both mathematics and physics literature. It is connected to topological entropy 

(see, e.g., [Fr], [G1], [G2], [DS]), and controlling it is necessary in order to construct 

interesting invariant measures and currents (see, e.g., [BF], [FS], [RS], [S]). Even in 

simple families of mappings, degree growth exhibits a rich behavior (see, e.g., the articles 

by Bedford and Kim [BK1], [BK2], which also contain references to the physics 

literature). 

In [FS], Fornaess and Sibony connected the degree growth of rational self-maps to 

the interplay between contracted hypersurfaces and indeterminacy points. In particular, 

they proved that deg(F n ) is multiplicative if and only if F is what is now often called 

(algebraically) stable. This analysis was extended to slightly more general maps in [N]. 

Bonifant and Fornaess [BF] showed that only countably many sequences (deg(F n )) ∞ 1 

can occur, but in general, the precise picture is unclear. 

For bimeromorphic maps of surfaces, the situation is quite well understood since 

the work of Diller and Favre [DF]. Using the factorization into blowups and blowdowns, 

they proved that any such map can be made stable by a bimeromorphic change 

of coordinates. This reduces the study of degree growth to the spectral properties of 

the induced map on the Dolbeault cohomology H 1,1 . In particular, it implies that λ 1 is 

an algebraic integer and that deg(F n ) satisfies an integral recursion formula and gives 

a stronger version of our main theorem when λ 2 1 > 1(= λ 2). 

In the case that we consider, namely, (noninvertible) meromorphic surface maps, 

there are counterexamples to stability when λ 2 1 = λ 2 > 1 (see [F]). It is an interesting 

(and probably difficult) question whether counterexamples also exist with 

λ 2 1 >λ 2 > 1. 

Instead of looking for a particular birational model in which the action of F n 

on H 1,1 can be controlled, we take a different tack and study the action of F on 

cohomology classes on all modifications π : X π → X at the same time. This idea 

already appeared in the study of cubic surfaces in [M] and was recently used by Cantat 

as a key tool in his investigation of the group of birational transformation of surfaces 

(see [C2]). In the context of noninvertible maps, Hubbard and Papadopol [HP] used 

similar ideas, but their methods apply only to a quite restricted class of maps. 

Here we show that F acts (functorially) by pullback F ∗ and pushforward F ∗ on 

the vector space W := lim H 1,1 

←− 

R 

(X π) and on its dense subspace C := lim H 1,1 

−→ 

R 

(X π). 

Compactness properties of W imply the existence of eigenvectors having eigenvalue 

λ 1 and certain positivity properties.

DEGREE GROWTH OF MEROMORPHIC SURFACE MAPS 521 

Following [DF], we then study the spectral properties of F ∗ and F ∗ under the 

assumption that λ 2 1 >λ 2. The space W is too big for this purpose, and we introduce a 

subspace L 2 that is the completion of C with respect to the (indefinite) inner product 

induced by the cup product, which is of Minkowski type by the Hodge index theorem. 

The main theorem then follows from the spectral properties of F ∗ and its adjoint F ∗ 

on L 2 . 

Using a different method, polynomial mappings of C 2 were studied in detail by 

Favre and Jonsson in [FJ4]: in that case, λ 1 is a quadratic integer. However, our main 

theorem for polynomial maps does not immediately follow from the analysis in [FJ4]; 

the methods of the two articles can be viewed as complementary. 

The space W above can be thought of as the Dolbeault cohomology H 1,1 of the 

Riemann-Zariski space of X. While we do not need the structure of the latter space 

in this article, the general philosophy of considering all bimeromorphic models at the 

same time is very useful for handling asymptotic problems in geometry, analysis, and 

dynamics (see [BFJ], [C1], [M], [FJ1], [FJ2], [FJ3]). In the present setting, it allows 

us to bypass the intricacies of indeterminacy points: heuristically, a meromorphic map 

becomes holomorphic on the Riemann-Zariski space. 

The article is organized in three sections. In the first, we recall some definitions 

and introduce cohomology classes on the Riemann-Zariski space. In the second, we 

study the actions of meromorphic mappings on these classes. Finally, Section 3 deals 

with the spectral properties of these actions under iteration, concluding with the proof 

of the main theorem. 

Remark on the setting. We choose to state our main result in the context of a complex 

manifold because the study of degree growth is particularly important for applications 

to holomorphic dynamics. However, our methods are purely algebraic, so that our main 

result actually holds in the case when X is a projective surface over any algebraically 

closed field of any characteristic, and ω = c 1 (L) for some ample line bundle. In this 

(X) with the real Néron-Severi vector space and work 

with the suitable notion of pseudoeffective and nef classes, as defined in [L, Sections 

1.4, 2.2]. 

context, one has to replace H 1,1 

R 

1. Classes on the Riemann-Zariski space 

Let X be a complex compact Kähler surface (for background, see [BHPV]), and write 

(X) := H 1,1 (X) ∩ H 2 (X, R). 

H 1,1 

R 

1.1. The Riemann-Zariski space 

By a blowup of X we mean a bimeromorphic morphism π : X π → X, where X π is a 

smooth surface. Up to isomorphism, π is then a finite composition of point blowups. 

If π and π ′ are two blowups of X, we say that π ′ dominates π and write π ′ ≥ π if 

there exists a bimeromorphic morphism µ : X π ′ → X π such that π ′ = π ◦ µ. The


Riemann-Zariski space of X is the projective limit 

X := lim ←−π 

X π . 

While suggestive, the space X is, strictly speaking, not needed for our analysis, and 

we refer to [ZS, Chapter 6, Section 17], [V, Section 7] for details on its structure. 

1.2. Weil and Cartier classes 

When one blowup π ′ = π ◦ µ dominates another one π, we have two induced linear 

maps, µ ∗ : H 1,1 

R (X 1,1 

π ′) → HR (X π) and µ ∗ : H 1,1 

R (X π) → H 1,1 

R 

(X π ′), which satisfy 

the projection formula µ ∗ µ ∗ = id. This allows us to define the following spaces. 


The space of Weil classes on X is the projective limit 

W (X) := lim ←−π 

H 1,1 

R (X π) 

with respect to the pushforward arrows. The space of Cartier classes on X is the 

inductive limit 

with respect to the pullback arrows. 

C(X) := lim −→π 

H 1,1 

R (X π) 

The space W (X) is endowed with its projective limit topology, that is, the coarsest 

topology for which the projection maps W (X) → H 1,1 

R 

(X π) are continuous. There is 

also an inductive limit topology on C(X), but we do not use it. 

Concretely, a Weil class α ∈ W (X) is given by its incarnations α π ∈ H 1,1 

R 

(X π), 

compatible by pushforward; that is, µ ∗ a π ′ = α π whenever π ′ = π ◦ µ. The topology 

on W (X) is characterized as follows: a sequence (or net † ) α j ∈ W(X) converges to 

α ∈ W (X) if and only if α j,π → α π in H 1,1 

R (X π) for each blowup π. 

The projection formula recalled above shows that there is an injection C(X) ⊂ 

W (X), so that a Cartier class is, in particular, a Weil class. In fact, if α ∈ H 1,1 

R 

(X π) is 

a class in some blowup X π of X,thenα defines a Cartier class, also denoted α, whose 

incarnation α π ′ in any blowup π ′ = π ◦ µ dominating π is given by α π ′ = µ ∗ α. 

We say that α is determined in X π . (It is then also determined in X π ′ for any blowup 

dominating π.) Each Cartier class is obtained that way. The space C(X) is dense in 

W (X): ifα is a given Weil class, the net α π of Cartier classes determined by the 

incarnations of α on all models X π tautologically converges to α in W(X). 

† A net is a family indexed by a directed set (see [Fo]).


Remark 1.2 

The spaces of Weil classes and Cartier classes are denoted Z • (X) and Z • (X) by 

Manin [M]. He views these classes as living on the bubble space lim X −→ π rather than 

the Riemann-Zariski space lim X ←− π . 

1.3. Exceptional divisors 

This section can be skipped on a first reading, the main technical issue being Proposition 

1.6, which is used for the proof of Theorem 3.2. 

The spaces C(X) and W (X) are clearly bimeromorphic invariants of X. Once 

the model X is fixed, an alternative and somewhat more explicit description of these 

spaces can be given in terms of exceptional divisors. 


The set D of exceptional primes over X is defined as the set of all exceptional prime 

divisors of all blowups X π → X modulo the following equivalence relation: two 

divisors E and E ′ on X π and X π ′ are equivalent if the induced meromorphic map 

X π X π ′ sends E onto E ′ . 

When X is a projective surface, D is the set of divisorial valuations on the function 

field C(X) whose center on X is a point. 

If E ∈ D is an exceptional prime and X π is any model of X, one can consider the 

center of E on X π , denoted by c π (E). It is a subvariety defined as follows. Choose 

ablowupπ ′ ≥ π so that E appears as a curve on X π ′.Thenc π (E) is defined as the 

image of E ⊂ X π ′ by the map X π ′ → X π . It does not depend on the choice of π ′ and 

is either a point or an irreducible curve. In this 2-dimensional setting, there is a unique 

minimal blowup π E such that c π (E) is a curve if and only if π ≥ π E . (In particular, 

c πE (E) is a curve.) 

Using these facts, one can construct an explicit basis for the vector space C(X) 

as follows (cf. [M, Proposition 35.6]). Let α E ∈ C(X) be the Cartier class determined 

by the class of E on X πE . Write R (D) for the direct sum ⊕ D 

R or, equivalently, for 

the space of real-valued functions on D with finite support. 


The set {α E | E ∈ D} is a basis for the vector space of Cartier classes α ∈ C(X) 

which are exceptional over X, that is, whose incarnations on X vanish. In other 

words, the map H 1,1 

R (X) ⊕ R(D) → C(X), sending α ∈ H 1,1 

R 

(X) to the Cartier class 

it determines, and E ∈ D to α E is an isomorphism. 

We now describe W (X) in terms of exceptional primes. If α ∈ W(X) is a given 

Weil class, let α X ∈ H 1,1 

R 

(X) be its incarnation on X. For each π, the Cartier class


α π − α X is determined on X π by a unique R-divisor Z π exceptional over X. IfE is 

a π-exceptional prime, we set ord E (α) := ord E (Z π ) so that Z π = ∑ E ord E(Z π )E. 

It is easily seen to depend only on the class of E in D. LetR D denote the (product) 

space of all real-valued functions on D. We obtain a map W(X) → H 1,1 

R (X) × RD , 

which is easily seen to be a bijection, and even naturally a homeomorphism, as the 

following straightforward lemma shows. 

LEMMA 1.5 

Anetα j ∈ W (X) converges to α ∈ W (X) if and only if α j,X converges to α X in 

H 1,1 

R (X) and ord E(α j ) → ord E (α) for each exceptional prime E ∈ D. 

A result of Zariski (cf. [Ko, Theorem 3.17], [FJ1, Proposition 1.12]) states that the 

process of successively blowing up the center of a given exceptional prime E ∈ D 

starting from any given model must stop after finitely many steps with the center 

becoming a curve. In other words, if X = X 0 ← X 1 ← X 2 ← ··· is an infinite 

sequence of blowups such that the center of each blowup X n ← X n+1 meets c Xn (E), 

then X n must dominate X πE for n large enough. Using this result, we record the 

following fact, which is used later on in the article. 


Let X = X 0 ← X 1 ← X 2 ← ··· be an infinite sequence of blowups, and for each 

n, suppose that α n ∈ C(X) is a Cartier class that is determined in X n+1 and whose 

incarnation on X n is zero. Then α n → 0 in W (X) as n →∞. 

Proof 

In view of Proposition 1.5, we have to show that for every given exceptional prime 

E ∈ D, ord E (α n ) converges to zero as n →∞. In fact, we claim that ord E (α n ) = 0 

for n ≥ n(E) large enough. Indeed, according to Zariski’s result, there are two 

possibilities: either there exists N such that c XN (E) is a curve, or there exists N such 

that the center of the blowup X n+1 → X n does not meet c Xn (E) for all n ≥ N.Inthe 

first case, it is clear that ord E (α n ) = 0 for n ≥ N since α n is exceptional over X N . 

In the second case, the center of E on X n does not meet the exceptional divisor of 

X n → X n−1 for n>N, which supports the exceptional class α n ; thus ord E (α n ) = 0 

for n>Nas well. 

 

1.4. Intersections and L 2 -classes 

For each π, the intersection pairing H 1,1 

R (X π)×H 1,1 

R (X π) → R is denoted by (α·β) Xπ . 

It is nondegenerate and satisfies the projection formula: (µ ∗ α · β) Xπ = (α · µ ∗ β) Xπ ′ if 

π ′ = π ◦ µ. It thus induces a pairing W (X) × C(X) → R which is denoted simply 

by (α · β).



The intersection pairing induces a topological isomorphism between W(X) and C(X) ∗ 

endowed with its weak-∗ topology. 

Proof 

A linear form L on C(X) = lim H 1,1 

−→π 

R 

(X π) is the same thing as a collection of 

linear forms L π on H 1,1 

R 

(X π), compatible by restriction. Now, such a collection is by 

definition an element of the projective limit lim H 1,1 

←−π 

R 

(X π) ∗ , which is identified to 

W (X) via the intersection pairing. This shows that the intersection pairing identifies 

W (X) with the dual of C(X) endowed with its weak-∗ topology. 

 

The intersection pairing defined above restricts to a nondegenerate quadratic form on 

C(X), denoted by α ↦→ (α 2 ). However, it does not extend to a continuous quadratic 

form on W (X). For instance, if z 1 ,z 2 ,...is a sequence of distinct points on X and π n 

denotes the blowup of X at z 1 ,...,z n , with exceptional divisor F n = E 1 +···+E n , 

we have (F 2 n ) =−n, but{F n}∈C(X) converges in W (X). We thus introduce the 

maximal space to which the intersection form extends. 


The space of L 2 -classes L 2 (X) is defined as the completion of C(X) with respect to 

the intersection form. 

The usual setting in which to perform a completion is that of a definite quadratic form 

on a vector space, which is not the case of the intersection form on C(X). However, 

the Hodge index theorem implies that it is of Minkowski type, and it is easy to show 

that the completion exists in that setting. 

Let us be more precise. If ω ∈ C(X) is a given class with (ω 2 ) > 0, the intersection 

form is negative definite on its orthogonal complement ω ⊥ :={α ∈ C(X) | (α·ω) = 0} 

(X π). Wehavean 

orthogonal decomposition C(X) = Rω ⊕ ω ⊥ , and we then let L 2 (X) := Rω ⊕ ω ⊥ , 

where ω ⊥ is the completion in the usual sense of ω ⊥ endowed with the negative 

definite quadratic form (α 2 ). Note that tω ⊕ α ↦→ t 2 − (α 2 ) is then a norm on L 2 (X) 

which makes it a Hilbert space, but this norm depends on the choice of ω. However, 

the topological vector space L 2 (X) does not depend on the choice of ω. 

In fact, the completion can be characterized by the following universal property: 

if (Y, q) is a complete topological vector space with a continuous nondegenerate 

quadratic form of Minkowski type, any isometry T : C(X) → Y continuously 

extends to L 2 (X) → Y . 

as a consequence of the Hodge index theorem applied to each H 1,1 

R


The intersection form on L 2 (X) is also of Minkowski type, so that it satisfies 

the Hodge index theorem: if a nonzero class α ∈ L 2 (X) satisfies (α 2 ) > 0, then the 

intersection form is negative definite on α ⊥ ⊂ L 2 (X). 

Remark 1.9 

The direct sum decomposition C(X) = H 1,1 

R 

(X) ⊕ R(D) of Proposition 1.4 is orthogonal 

with respect to the intersection form. Furthermore, the intersection form is negative 

definite on R (D) ,and{α E | E ∈ D} forms an orthonormal basis for −(α 2 ).Indeed,the 

center of E ∈ D on the minimal model X πE on which it appears is necessarily the last 

exceptional divisor to have been created in any factorization of π E into a sequence of 

point blowups; thus it is a (−1)-curve. 

Using this, one sees that L 2 (X) is isomorphic to the direct sum H 1,1 

R 

(X)⊕l2 (D) ⊂ 

W (X), where l 2 (D) denotes the set of real-valued, square-summable functions E ↦→ 

a E on D. 

The different spaces that we have introduced so far are related as follows. 


There is a natural continuous injection L 2 (X) → W (X), and the topology on L 2 (X) 

induced by the topology of W (X) coincides with its weak topology as a Hilbert space. 

If α ∈ W (X) is a given Weil class, then the intersection number (απ 2 ) is a 

decreasing function of π, and α ∈ L 2 (X) if and only if (απ 2 ) is bounded from below, 

in which case, (α 2 ) = lim π (απ 2 ). 

Proof 

The injection L 2 (X) → W(X) is dual to the dense injection C(X) ⊂ L 2 (X). By 

Proposition 1.7,anetα k ∈ L 2 (X) converges to α ∈ L 2 (X) in the topology induced by 

W (X) if and only if (α k · β) → (α · β) for each β ∈ C(X). SinceC(X) is dense in 

L 2 (X), thisimpliesthatα k → α weakly in L 2 (X). 

For the last part, one can proceed using the abstract definition of L 2 (X) as a 

completion, but it is more transparent to use the explicit representation of Remark 1.9. 

For any π, wehaveα π = α X + ∑ E∈D π 

(α · α E )α E , where D π ⊂ D is the set of 

exceptional primes of π. Then(απ 2 ) = (α2 X ) − ∑ E∈D π 

(α · α E ) 2 , which is decreasing 

in π. It is then clear that α ∈ L 2 (X) if and only if (απ 2 ) is uniformly bounded from 

below and (α 2 ) = lim(απ 2 ). 

 

1.5. Positivity 

Recall that a class in H 1,1 

R 

(X) is pseudoeffective (psef) if it is the class of a closed 

positive (1, 1)-current on X. It is numerically effective (nef) if it is the limit of Kähler


classes. Any nef class is psef. The cone in H 1,1 

R 

(X) consisting of psef classes is strict: 

if α and −α are both psef, then α = 0. 

If π ′ = π ◦ µ is a blowup dominating some other blowup π,thenα ∈ H 1,1 

R (X π) 

is psef (nef) if and only if µ ∗ α ∈ H 1,1 

R 

(X π ′) is psef (nef ). On the other hand, if 

α ′ ∈ H 1,1 

R (X π ′) is psef (nef ), then so is µ ∗α ′ ∈ H 1,1 

R 

(X π). (For the nef part of the last 

assertion, it is important that we work in dimension two.) 


AWeilclassα ∈ W (X) is psef (nef ) if its incarnation α π ∈ H 1,1 

R (X π) is psef (nef ) 

for any blowup π : X π → X. 

We denote by Nef(X) ⊂ Psef(X) ⊂ W (X) the convex cones of nef and psef classes. 

The remarks above imply that a Cartier class α ∈ C(X) is psef (nef ) if and only if 

α π ∈ H 1,1 

R 

(X π) is psef (nef ) for one (or any) X π in which α is determined. We write 

α ≥ β as a shorthand for α − β ∈ W (X) being psef. 


The nef cone Nef(X) and the psef cone Psef(X) are strict, closed, convex cones in 

W (X) with compact bases. 

Proof 

The nef (resp., psef ) cone is the projective limit of the nef (resp., psef ) cones of each 

H 1,1 

R 

(X π). These are strict, closed, convex cones with compact bases, so the result 

follows from the Tychonoff theorem. 

 

Nef classes satisfy the following monotonicity property. 


If α ∈ W (X) is a nef Weil class, then α ≤ α π for each π. In particular, α π ≠ 0 for 

each π unless α = 0. 

Proof 

By induction on the number of blowups, it suffices to prove that α π ′ ≤ µ ∗ α π when 

π ′ = π ◦ µ and µ is the blowup of a point in X π .Butthenµ ∗ α π = α π ′ + cE, where 

E is the class of the exceptional divisor and c = (α π ′ · E) ≥ 0. To get the second 

point, note that α π = 0 for some π implies that α ≤ 0. On the other hand, α ≥ 0 as 

α is nef. Since Psef(X) is a strict cone, we infer that α = 0. 

 


The nef cone Nef(X) is contained in L 2 (X).Ifα i ≥ β i , i = 1, 2, are nef classes, then 

we have (α 1 · α 2 ) ≥ (β 1 · β 2 ) ≥ 0.


Proof 

If α ∈ W (X) is nef, each incarnation α π is nef, and thus (απ 2 ) ≥ 0,sothatα ∈ L2 (X) 

by Proposition 1.10 with (α 2 ) = inf π (απ 2 ) ≥ 0. To get the second point, note that 

(α 1 · α 2 ) ≥ (α 1 · β 2 ) since α 2 − β 2 is psef and α 1 is nef and, similarly, (α 1 · β 2 ) ≥ 

(β 1 · β 2 ). 

These two propositions together show that if ω ∈ C(X) is a Cartier class determined 

by a Kähler class down on X,then(α · ω) > 0 for any nonzero nef class α ∈ W(X). 


We have 2(α · β) α ≥ (α 2 ) β for any nef Weil classes α, β ∈ W(X). In particular, if 

ω ∈ C(X) is determined by a Kähler class on X normalized by (ω 2 ) = 1, we have, 

for any nonzero nef Weil class α, 

(α 2 ) 

ω ≤ α ≤ 2(α · ω) ω. (1.1) 

2(α · ω) 

Proof 

The second assertion is a special case of the first one. To prove the first one, we may 

assume that (α · β) > 0,orelseα and β are proportional by the Hodge index theorem, 

and the result is clear. It is a known fact (see the remark after [B, Theorem 4.1]) that 

if γ ∈ C(X) is a Cartier class with (γ 2 ) ≥ 0, then either γ or −γ is psef. In view of 

Proposition 1.10, the same result is true for any γ ∈ L 2 (X). Apply this to γ = α − tβ, 

where t = ((α · α)/2(α · β)).As(γ · γ ) ≥ 0 and (γ · α) ≥ 0, γ must be psef. 

1.6. The canonical class 

The canonical class K X is the Weil class whose incarnation in any blowup X π is the 

canonical class K Xπ . It is not Cartier and does not even belong to L 2 (X). However, 

K Xπ ′ ≥ K Xπ whenever π ′ ≥ π, andK X is the smallest Weil class dominating all the 

K Xπ . This allows us to intersect K X with any nef Weil class α in a slightly ad hoc 

way: we set (α · K X ):= sup π (α π · K Xπ ) Xπ ∈ R ∪{+∞}. 

2. Functorial behavior 

Throughout this section, let F : X Y be a dominant meromorphic map between 

compact Kähler surfaces. Following [M, Section 34.7], we introduce the action of F 

on Weil and Cartier classes. We then describe the continuity properties of these actions 

on the Hilbert space L 2 (X). 

For each blowup Y ϖ of Y , there exists a blowup X π of X such that the induced 

map X π → Y ϖ is holomorphic. The associated pushforward H 1,1 

R (X π) → H 1,1 

R (Y ϖ ) 

and pullback H 1,1 

R (Y ϖ ) → H 1,1 

R 

(X π) are compatible with the projective and injective 

systems defined by pushforwards and pullbacks that define Weil and Cartier classes,


respectively, so we can consider the induced morphisms on the respective projective 

and inductive limits. 


Given F : X Y as above, we denote by F ∗ : W (X) → W(Y) the induced 

pushforward operator and by F ∗ : C(Y) → C(X) the induced pullback operator. 

Concretely, if α ∈ W (X) is a Weil class, the incarnation of F ∗ α ∈ W(Y) on a given 

blowup Y ϖ is the pushforward of α π ∈ H 1,1 

R 

(X π) by the induced map X π → Y ϖ for 

any π such that the latter map is holomorphic. Similarly, if β ∈ C(Y) is a Cartier class 

determined on a blowup Y ϖ , its pullback F ∗ β ∈ C(X) is the Cartier class determined 

on X π by the pullback of β ϖ ∈ H 1,1 

R 

(Y ϖ ) by the induced map X π → Y ϖ , whenever 

the latter is holomorphic. 

These constructions are functorial, that is, (F ◦ G) ∗ = F ∗ ◦ G ∗ and (F ◦ G) ∗ = 

G ∗ ◦ F ∗ , and are compatible with the duality between C and W since this is true for 

each holomorphic map X π → Y ϖ . In other words, for any α ∈ W(X) and β ∈ C(Y), 

we have (F ∗ α · β) = (α · F ∗ β). 

We also see that F ∗ preserves nef and psef Weil classes and that F ∗ preserves 

nef and psef Cartier classes. Indeed, the pullback and pushforward by a surjective 

holomorphic map both preserve nef and psef (1, 1)-classes in dimension two. 

Remark 2.2 

If π : X π → X and ϖ : Y ϖ → Y are arbitrary blowups, then the pullback operator 

H 1,1 

R (Y ϖ ) → H 1,1 

R (X π) usually associated to the meromorphic map X π Y ϖ is 

(Y ϖ ), followed by the projection 

of C(X) onto H 1,1 

R (X π). Similarly, the pushforward operator H 1,1 

R (X π) → H 1,1 

R (Y ϖ ) 

usually associated to X π Y ϖ is given by the restriction of F ∗ : W(X) → W(Y) 

given by the restriction of F ∗ : C(Y) → C(X) to H 1,1 

R 

to H 1,1 

R (X π), followed by the projection of W (Y) onto H 1,1 

R (Y ϖ ). 

The intersection forms on C(X) and C(Y) are related by F ∗ as follows: (F ∗ β 2 ) = 

e(F )(β 2 ), where e(F ) > 0 is the topological degree of F . In view of the universal 

property of completions mentioned in Section 1.4, we get the following. 


The pullback F ∗ : C(Y) → C(X) extends to a continuous operator F ∗ :L 2 (Y) → 

L 2 (X), so that ((F ∗ β) 2 ) = e(F )(β 2 ) for each β ∈ L 2 (Y). By duality, the pushforward 

F ∗ : W (X) → W (Y) induces a continuous operator F ∗ :L 2 (X) → L 2 (Y), so that 

(F ∗ α · β) = (α · F ∗ β) for any α, β ∈ L 2 (X). 

Next, we show that the pullback F ∗ : C(Y) → C(X) continuously extends to Weil 

classes and—dually—that the pushforward F ∗ : W (X) → W(Y) preserves Cartier 

classes.


In doing so, we repeatedly use a consequence of the result of Zariski mentioned 

in Section 1. Namely, given F : X Y and a blowup π : X π → X, there exists 

ablowupY ϖ of Y such that the induced meromorphic map X π Y ϖ does not 

contract any curve to a point. 

LEMMA 2.4 

Suppose that π : X π → X and ϖ : Y ϖ → Y are two blowups such that the induced 

meromorphic map X π Y ϖ does not contract any curve to a point. Then for each 

Cartier class β ∈ C(Y), the incarnations of F ∗ β and F ∗ β ϖ on X π coincide. 

Proof 

Any Cartier class is a difference of nef Cartier classes, so we may assume that β is 

nef and determined in some blowup ϖ ′ dominating ϖ .Pickπ ′ dominating π so that 

the induced map X π ′ → Y ϖ ′ is holomorphic. Set α := F ∗ (β ϖ − β).Thenα ∈ C(X) 

is psef and determined in X π ′. We must show that α π = 0. Ifα π ≠ 0, thenα ≥ λC, 

where λ>0 and C is the class of an irreducible curve on X π .Now,C is not contracted 

by X π Y ϖ , so the incarnation of F ∗ α on Y ϖ is nonzero. But this is a contradiction 

since this incarnation equals e(F )(β ϖ − β) ϖ = 0. 

 

COROLLARY 2.5 

The pullback operator F ∗ : C(Y) → C(X) continuously extends to F ∗ : W(Y) → 

W (X) and preserves nef and psef Weil classes. 

More precisely, if X π is a given blowup of X and Y ϖ is a blowup of Y such that 

the induced meromorphic map X π Y ϖ does not contract curves, then for any Weil 

class γ ∈ W (Y), one has (F ∗ γ ) π = (F ∗ γ ϖ ) π . 

COROLLARY 2.6 

The pushforward operator F ∗ : W (X) → W (Y) preserves Cartier classes. More 

precisely, if α ∈ C(X) is a Cartier class determined on some X π ,thenF ∗ α is Cartier, 

determined on Y ϖ as soon as the induced meromorphic map X π Y ϖ does not 

contract curves. 

Proof 

For any β ∈ C(Y), the incarnations of F ∗ β and F ∗ β ϖ on X π coincide by Corollary 

2.5. Hence 

(F ∗ α · β) = (α · F ∗ β) = (α · F ∗ β ϖ ) = (F ∗ α · β ϖ ) = ( (F ∗ α) ϖ · β ) . 

As this holds for any Cartier class β ∈ C(Y), we must have F ∗ α = (F ∗ α) ϖ 

Proposition 1.7. 

by


3. Dynamics 

Now, consider a dominant meromorphic self-map F : X X of a compact Kähler 

surface X. Write λ 2 = e(F ) for the topological degree of F .Ifω ∈ Nef(X) is a nef 

Weil class such that (ω 2 ) > 0, we define the degree of F with respect to ω as 

deg ω (F ):= (F ∗ ω · ω) = (ω · F ∗ ω). 

This coincides with the usual notion of degree when X = P 2 and ω is the Cartier class 

determined by a line on P 2 . 


The limit 

λ 1 := λ 1 (F ):= lim 

n→∞ 

deg ω (F n ) 1/n (3.1) 

exists and does not depend on the choice of the nef class ω ∈ Nef(X) with (ω 2 ) > 0. 

Moreover, λ 1 is invariant under bimeromorphic conjugacy and λ 2 1 ≥ λ 2. 

The result above is well known, but we include the proof for completeness. We call 

λ 1 the asymptotic degree of F .Itisalsoknownasthefirst dynamical degree and can 

be computed (see [DF]) as λ 1 = lim n→∞ ρn 

1/n , where ρ n is the spectral radius of F n 

acting on H 1,1 

R 

(X) by pullback or pushforward (cf. Remark 2.2). 

Proof of Proposition 3.1 

Upon scaling ω, we can assume that (ω 2 ) = 1. By(1.1), we then have G ∗ ω ≤ 

2(G ∗ ω · ω) ω for any dominant mapping G : X X. Applying this with G = F m 

yields 

deg ω (F n+m ) = (F n∗ F m∗ ω · ω) ≤ 2(F n∗ ω · ω)(F m∗ ω · ω) = 2deg ω (F n )deg ω (F m ). 

This implies (see, e.g., [KH, Proposition 9.6.4]) that the limit in (3.1) exists. Let us 

temporarily denote it by λ 1 (ω).Ifω ′ ∈ C(X) is another nef class with (ω ′ 2 ) > 0,then 

it follows from (1.1) thatω ′ ≤ Cω for some C>0. By Proposition 1.14, thisgives 

deg ω ′(F n ) = (F n∗ ω ′ · ω ′ ) ≤ C 2 (F n∗ ω · ω) = C 2 deg ω (F n ). 

Taking nth roots and letting n →∞shows that λ 1 (ω ′ ) ≤ λ 1 (ω), and thus λ 1 (ω ′ ) = 

λ 1 (ω) by symmetry, so that λ 1 is indeed independent of ω. It is then invariant by 

bimeromorphic conjugacy since X and all the spaces attached to it are. 

Finally, Proposition 1.14 yields F n∗ ω ≤ 2(F ∗n ω · ω) ω, which implies that 

e(F ) n = e(F n ) = (F n∗ ω 2 ) ≤ 4(F n∗ ω · ω) 2 = 4deg ω (F n ) 2 , 

and letting n →∞yields λ 2 = e(F ) ≤ λ 2 1 .


3.1. Existence of eigenclasses 

To begin, we do not assume that λ 2 1 >λ 2. 

THEOREM 3.2 

Let F : X X be any dominant meromorphic self-map of a smooth Kähler surface 

X with asymptotic degree λ 1 . Then we can find nonzero nef Weil classes θ ∗ and θ ∗ 

with F ∗ θ ∗ = λ 1 θ ∗ and F ∗ θ ∗ = λ 1 θ ∗ . 

Note that by Proposition 1.14, both classes θ ∗ ,θ ∗ belong to L 2 (X). 

Proof 

We use the pushforward and pullback operators 

S π : H 1,1 

R (X π) → H 1,1 

R (X π) and T π : H 1,1 

R (X π) → H 1,1 

R 

(X π), 

usually associated to the meromorphic map X π X π induced by F for a given 

blowup π : X π → X. Thus S π (resp., T π ) is the restriction to H 1,1 

R (X π) of F ∗ : 

C(X) → C(X) (resp., F ∗ : C(X) → C(X)) followed by the projection C(X) → 

H 1,1 

R 

(X π) (cf. Remark 2.2). These operators are typically denoted F ∗ and F ∗ in the 

literature, but here that notation conflicts with the corresponding operators on C(X) 

or W (X). 

The spectral radius ρ π > 0 of T π can be computed as follows: if θ ∈ H 1,1 

R (X π) 

is any nef class with (θ 2 ) > 0, then(Tπ nθ · θ)1/n → ρ π as n →∞. 

LEMMA 3.3 

We have λ 1 ≤ ρ π ′ ≤ ρ π for all π ′ ≥ π. 

Proof 

Let θ ∈ C(X) be a given nef class determined on X π ′ with (θ 2 ) > 0,sothatθ ≤ θ π by 

Proposition 1.13.ThenT π ′θ is the incarnation on X π ′ of the nef class F ∗ θ on X π ′,and 

T π θ π is the incarnation on X π of the nef class F ∗ θ π ≥ F ∗ θ; thus F ∗ θ ≤ T π ′θ ≤ T π θ π 

holds by Proposition 1.13. By induction, we get F n∗ θ ≤ Tπ n ′θ ≤ T π nθ π for all n; hence 

(F n∗ θ ·θ) 1/n ≤ (Tπ n ′θ ·θ)1/n ≤ (Tπ nθ π ·θ π ) 1/n by Proposition 1.14,andλ 1 ≤ ρ π ′ ≤ ρ π 

follows by letting n →∞. 

 

Now, the set of nef classes in H 1,1 

R 

(X π) is a closed convex cone with compact basis 

invariant by T π ; thus a Perron-Frobenius-type argument (see [DF, Lemma 1.12]) 

establishes the existence of a nonzero nef class θ(π) ∈ H 1,1 

R (X π) with T π θ(π) = 

ρ π θ(π).


If we identify θ(π) with the nef Cartier class that it determines, this says that the 

nef Cartier classes F ∗ θ(π) and ρ π θ(π) have the same incarnation on X π .Wehave 

thus obtained approximate eigenclasses, and now the plan is to get the desired class 

θ ∗ as a limit of classes of the form θ(π). We then explain how to modify the argument 

to construct θ ∗ . 

We normalize θ(π) by (θ(π) · ω) = 1 for a fixed class ω ∈ C(X) determined by 

aKähler class on X with (ω 2 ) = 1, so that the θ(π) all lie in a compact subset of the 

nef cone Nef(X) by Proposition 1.12. 

Let X = X 0 ← X 1 ← ··· be an infinite sequence of blowups so that the lift of 

F as a map from X n+1 to X n is holomorphic for n ≥ 0. 

For each n, letρ n denote the spectral radius of T n on H 1,1 

R 

(X n) as above, 

and pick a nonzero nef Cartier class θ n ∈ C(X) determined on X n and such that 

T n θ n = ρ n θ n .ThenF ∗ θ n is a Cartier class determined in X n+1 , and by definition, 

T n θ n is the incarnation of this class in X n . Therefore F ∗ θ n and ρ n θ n coincide on 

X n . By Proposition 1.6, it follows that F ∗ θ n − ρ n θ n converges to zero in W(X) as 

n →∞. 

We have seen above that ρ n is a decreasing sequence. Let ρ ∞ := lim ρ n ,sothat 

ρ ∞ ≥ λ 1 by Lemma 3.3. Since the θ n lie in a compact subset of Nef(X), we can find 

a cluster point θ ∗ for the sequence θ n , which is also a nef Weil class with (θ ∗ · ω) = 1. 

Since F ∗ θ n − ρ n θ n converges to zero in W (X), it follows that F ∗ θ ∗ = ρ ∞ θ ∗ . 

To complete the proof, we show that ρ ∞ = λ 1 . In fact, if α ∈ W(X) is any 

nonzero nef eigenclass of F ∗ with F ∗ α = tαfor some t ≥ 0,thent ≤ λ 1 .Indeed,we 

have α ≤ Cω for some C>0 by Proposition 1.15, and it follows that (F n∗ ω · ω) ≥ 

C −1 (F n∗ α · ω) = C −1 t n (α · ω). Takingnth roots and letting n →∞yields λ 1 ≥ t. 

In order to construct θ ∗ , we modify the above argument as follows. Let S π : 

H 1,1 

R (X π) → H 1,1 

R 

(X π) be the pushforward operator defined above. As F ∗ and F ∗ are 

adjoint to each other with respect to the intersection pairing, it follows that S π and T π 

are adjoint with respect to Poincaré duality on H 1,1 

R 

(X π), so that they have the same 

spectral radius ρ π . By a Perron-Frobenius-type argument, there exists a nonzero nef 

class ϑ(π) ∈ H 1,1 

R 

(X π) such that S π ϑ(π) = ρ π ϑ(π). 

Now, pick X = X 0 ← X 1 ←··· to be an infinite sequence of blowups such that 

the lifts of F from X n to X n+1 do not contract any curves. For each n, we get a nef class 

ϑ n ∈ C(X) determined on X n normalized by (ϑ n · ω) = 1. By Corollary 2.6, the class 

F ∗ ϑ n is determined in X n+1 ,soF ∗ ϑ n and ρ n ϑ n coincide in X n . Proposition 1.6 then 

shows that F ∗ ϑ n − ρ n ϑ n converges to zero in W (X) as n →∞; hence θ ∗ ∈ Nef(X) 

can be taken to be any cluster value of ϑ n . 

 

Remark 3.4 

When K X is not psef (i.e., if X is rational or ruled) we may also achieve (θ ∗ ·K X ) ≤ 0. 

To see this, first note that F ∗ K X ≤ K X as classes in W(X) since K Xπ ′ − F ∗ K Xπ


is represented by the effective zero divisor of the Jacobian determinant of the map 

X π ′ → X π induced by F , assuming that this is holomorphic. Now, for each blowup 

X π ,letC π be the set of nef classes α ∈ H 1,1 

R 

(X π) such that (α · K X ) ≤ 0.ThenC π is 

a closed convex cone with compact basis and is not reduced to zero since K X is not 

psef. It is, furthermore, invariant by S π . Indeed, if α ∈ H 1,1 

R 

(X π) is a nef class, we 

have 

(S π α · K X ) = (F ∗ α · K Xπ ) ≤ (F ∗ α · K X ) = (α · F ∗ K X ) ≤ (α · K X ). 

We can thus assume that the nonzero eigenclasses ϑ n in the proof of Theorem 3.2 

belong to C n , and we get (θ ∗ · K X ) ≤ 0. 

The same argument does not work for θ ∗ since F ∗ K X ≤ K X does not hold in 

general. 

3.2. Spectral properties 

Theorem 3.2 asserts the existence of eigenclasses for F ∗ and F ∗ with eigenvalue λ 1 . 

We now further analyze the spectral properties under the assumption that λ 2 1 >λ 2. 

THEOREM 3.5 

Assume that λ 2 1 >λ 2. Then the nonzero nef Weil classes θ ∗ ,θ ∗ ∈ L 2 (X) such that 

F ∗ θ ∗ = λ 1 θ ∗ and F ∗ θ ∗ = λ 1 θ ∗ are unique up to scaling. We have (θ ∗ · θ ∗ ) > 0 and 

(θ ∗2 ) = 0. We rescale them so that (θ ∗ · θ ∗ ) = 1. LetH ⊂ L 2 (X) be the orthogonal 

complement of θ ∗ and θ ∗ , so that we have the decomposition L 2 (X) = Rθ ∗ ⊕Rθ ∗ ⊕H. 

The intersection form is negative definite on H, and ‖α‖ 2 :=−(α 2 ) defines a Hilbert 

norm on H. The actions of F ∗ and F ∗ with respect to this decomposition are as 

follows. 

(i) The subspace H is F ∗ -invariant, and 

⎧ 

F n∗ θ ∗ = λ n 1 θ ∗ , 

( ⎪⎨ λ2 

) ( 

nθ∗ 

F n∗ θ ∗ = + (θ 2 ∗ 

λ ) λn 1 

1 − 

1 

with h ⎪⎩ 

n ∈ H, ‖h n ‖=O(λ n/2 

2 ), 

‖F n∗ h‖=λ n/2 

2 ‖h‖ for all h ∈ H. 

( λ2 

λ 2 1 

) n 

) 

θ ∗ + h n 

(ii) The subspace H is not F ∗ -invariant in general, but 

⎧ 

F∗ ⎪⎨ 

nθ ∗ = λ n 1 θ ∗, 

( 

F∗ nθ λ2 

) nθ ∗ = 

∗ , 

λ 1 

⎪⎩ ‖F∗ nh‖≤Cλn/2 

2 ‖h‖ for some C>0 and all h ∈ H.


COROLLARY 3.6 

For any Weil class α ∈ L 2 (X), we have 

( 

1 

(λ2 ) ) n/2 

F n∗ α = (α · θ ∗ )θ ∗ + O 

λ n 1 

λ 2 1 

and 

( 

1 

(λ2 ) ) n/2 

F n ∗ α = (α · θ ∗ )θ ∗ + O . 

λ n 1 

λ 2 1 

Proof 

The decomposition of α in L 2 (X) = Rθ ∗ ⊕ Rθ ∗ ⊕ H is given by 

α = ( (α · θ ∗ ) − (α · θ ∗ )(θ 2 ∗ )) θ ∗ + (α · θ ∗ )θ ∗ + α 0 , (3.2) 

where α 0 ∈ H. The result follows from (3.2) using Theorem 3.5(i), (ii). 

 

Proof of the main theorem 

Applying Corollary 3.6 to α = ω (which is nef and hence in L 2 (X))gives 

deg ω (F n ) = (F n∗ ω · ω) = (ω · θ ∗ )(ω · θ ∗ )λ n 1 + O(λn/2 2 ). 

This completes the proof with b := (ω · θ ∗ )(ω · θ ∗ ). 

 

Proof of Theorem 3.5 

Using Theorem 3.2, we may find nonzero nef Weil classes θ ∗ ,θ ∗ such that F ∗ θ ∗ = λ 1 θ ∗ 

and F ∗ θ ∗ = λ 1 θ ∗ . Fix two such classes for the duration of the proof. In the end, we 

see that they are unique up to scaling. 

The proof amounts to a series of simple arguments using general facts for transformations 

of a complete vector space endowed with a Minkowski form. We provide 

the details for the benefit of the reader. 

First, note that λ 1 F ∗ θ ∗ = F ∗ F ∗ θ ∗ = λ 2 θ ∗ ,sothatF ∗ θ ∗ = (λ 2 /λ 1 )θ ∗ .Since 

F ∗ θ ∗ = λ 1 θ ∗ and λ 2 1 >λ 2, it follows that θ ∗ and θ ∗ cannot be proportional. 

Applying the relation (F ∗ α 2 ) = λ 2 (α 2 ) to α = θ ∗ yields λ 2 1 (θ ∗2 ) = λ 2 (θ ∗2 ),and 

thus (θ ∗2 ) = 0 since λ 2 1 >λ 2. By the Hodge index theorem, θ ∗ and θ ∗ would thus 

have to be proportional if they were orthogonal. We infer that (θ ∗ · θ ∗ ) > 0, andwe 

rescale θ ∗ so that (θ ∗ · θ ∗ ) = 1. 

Let us first prove the properties in (i) for the pullback. As both θ ∗ and θ ∗ are 

eigenvectors for F ∗ , the space H is invariant under F ∗ . Using (3.2) and the invariance 

properties of θ ∗ and θ ∗ ,weget 

F ∗ θ ∗ = λ 2 

λ 1 

θ ∗ + λ 1 

( 

1 − λ 2 

λ 2 1 

) 

(θ 2 ∗ )θ ∗ + h 1 , (3.3)


where h 1 ∈ H. Inductively, (3.3)gives 

F n∗ θ ∗ = 

( λ2 

λ 1 

) nθ∗ 

+ λ n 1 

( 

1 − 

( λ2 

λ 2 1 

) n 

) 

(θ 2 ∗ )θ ∗ + h n , (3.4) 

where h n+1 = F ∗ h n + (λ 2 /λ 1 ) n h 1 ∈ H. Using the fact that ‖F ∗ h‖ 2 = λ 2 ‖h‖ 2 on 

H, weget‖h n+1 ‖≤λ 1/2 

2 ‖h n ‖+(λ 2 /λ 1 ) n ‖h 1 ‖, which is easily seen to imply that 

‖h n ‖=O(λ n/2 

2 ) since ∑ k (λ1/2 2 /λ 1 ) k < +∞. This concludes the proof of (i). 

Let us now turn to the pushforward operator. The first two equations are clear. As 

θ ∗ may not be an eigenvector for F ∗ , H need not be invariant by F ∗ , but since F ∗ h is 

orthogonal to θ ∗ for any h ∈ H, we can write F∗ nh = a nθ ∗ +g n with a n = (F n∗ θ ∗ ·h) 

and g n ∈ H. We have seen that F n∗ θ ∗ = h n modulo θ ∗ ,θ ∗ with ‖h n ‖=O(λ n/2 

2 ); 

thus |a n |=|(h n · h)| ≤Cλ n/2 

2 ‖h‖. On the other hand, we have (gn 2) = (F n∗ g n · h); 

thus ‖g n ‖ 2 ≤ λ n/2 

2 ‖g n ‖‖h‖, and this shows that ‖F∗ nh‖≤Cλn/2 

2 ‖h‖. 

Remark 3.7 

It follows from the proof of the main theorem that there exist nef classes α ∗ ,α ∗ ∈ 

H 1,1 

R 

(X) such that for any Kähler classes ω, ω′ on X, wehave 

( 

deg ω (F n ) 

deg ω ′(F n ) = (α∗ · ω) X (α ∗ · ω) (λ2 ) ) n/2 X 

+ O . 

(α ∗ · ω ′ ) X (α ∗ · ω ′ ) X 

Indeed, we can take α ∗ and α ∗ as the incarnations in X of θ ∗ and θ ∗ , respectively. 

Remark 3.8 

When F is bimeromorphic, we have θ ∗ (F ) = θ ∗ (F −1 ); hence (θ∗ 2 ) = 0. However,in 

general, we may have (θ∗ 2 ) > 0. For example, let F be any polynomial map of C2 

whose extension to P 2 is not holomorphic but does not contract any curve. If ω is the 

class of a line on P 2 ,thendeg ω (F ) > √ λ 2 > 1. On the other hand, F ∗ ω = deg ω (F )ω 

by Corollary 2.6,soλ 1 = deg ω (F ), θ ∗ = ω and (θ∗ 2) = 1. 

Remark 3.9 

The case when θ ∗ (or θ ∗ ) is Cartier is very special. For example, when F is bimeromorphic, 

it follows from [DF, Theorem 0.4] that θ ∗ (or, equivalently, θ ∗ ) is Cartier if 

and only if F is biholomorphic in some birational model. In the general noninvertible 

case, similar rigidity results are expected (see [C1] for work in this direction). 

Note also that F being algebraically stable in some birational model does not 

imply that the eigenclasses are Cartier. We do not know whether having a Cartier 

eigenclass implies algebraic stability in some model, but having a Cartier eigenclass 

has many of the same consequences as stability: λ 1 is an algebraic integer, and the 

sequence of degrees (deg ω F n ) ∞ 1 satisfies a linear recurrence relation. 

λ 2 1


Acknowledgments. We thank Serge Cantat and Jeff Diller for many useful remarks 

and the referees for careful readings of the article. 

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Boucksom 

Institut de Mathématiques, CNRS-Université Paris 7, F-75251 Paris CEDEX 05, France; 

boucksom@math.jussieu.fr 

Favre 

Institut de Mathématiques, CNRS-Université Paris 7, F-75251 Paris CEDEX 05, France; 

favre@math.jussieu.fr 

Jonsson 

Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109, USA; 

mattiasj@umich.edu

DISTORTION OF HAUSDORFF MEASURES AND 

IMPROVED PAINLEVÉ REMOVABILITY FOR 

QUASIREGULAR MAPPINGS 

K. ASTALA, A. CLOP, J. MATEU, J. OROBITG, and I. URIARTE-TUERO 

Abstract 

The classical Painlevé theorem tells us that sets of zero length are removable for 

bounded analytic functions, while (some) sets of positive length are not. For general 

K-quasiregular mappings in planar domains, the corresponding critical dimension is 

2/(K + 1). We show that when K>1, unexpectedly one has improved removability. 

More precisely, we prove that sets E of σ -finite Hausdorff (2/(K + 1))-measure are 

removable for bounded K-quasiregular mappings. On the other hand, dim(E) = 

2/(K + 1) is not enough to guarantee this property. 

We also study absolute continuity properties of pullbacks of Hausdorff measures 

under K-quasiconformal mappings: in particular, at the relevant dimensions 

1 and 2/(K + 1). For general Hausdorff measures H t , 0 < t < 2, we reduce 

the absolute continuity properties to an open question on conformal mappings (see 

Conjecture 2.3). 


A homeomorphism φ : → ′ between planar domains , ′ ⊂ C is called K- 

quasiconformal if it belongs to the Sobolev space W 1,2 

loc () and satisfies the distortion 

inequality 

max |∂ αφ| ≤K min|∂ α φ| a.e. in . (1.1) 

α 

α 

It has been known since the work of Ahlfors [3] that quasiconformal mappings preserve 

sets of zero Lebesgue measure. It is also well known that they preserve sets of zero 

Hausdorff dimension since K-quasiconformal mappings are Hölder continuous with 

exponent 1/K (see Mori [22]). However, these maps do not preserve Hausdorff 


Vol. 141, No. 3, c○ 2008 DOI 10.1215/00127094-2007-005 

Received 28 July 2006. Revision received 13 April 2007. 

2000 Mathematics Subject Classification. Primary 30C62; Secondary 35J15, 35J70. 

Astala supported in part by Academy of Finland projects 106257, 110641, and 211485. 

Clop supported in part by European Union project Conformal Structures and Dynamics (CODY). 

Clop, Mateu, and Orobitg supported in part by projects MTM2004-00519 (Spain), Acción Integrada HF2004- 

0208 (Spain), and 2005-SGR-00774 (Generalitat de Catalunya). 

Uriarte-Tuero supported by Academy of Finland projects 209371 and 203949. 

539

540 ASTALA, CLOP, MATEU, OROBITG, and URIARTE-TUERO 

dimension in general, and it was in the article of Astala [4] where the precise bounds 

for the distortion of dimension were given. For any compact set E with dimension t 

and for any K-quasiconformal mapping φ, wehave 

1 

( 1 

K t − 1 ) 

≤ 

2 

1 

dim(φ(E)) − 1 ( 1 

2 ≤ K t − 1 ) 

. (1.2) 

2 

Furthermore, these bounds are optimal (i.e., equality may occur in either estimate). 

The fundamental question that we study in this article is whether the estimates 

(1.2) can be improved to the level of Hausdorff measures H t . In other words, if φ is 

aplanarK-quasiconformal mapping, 0

DISTORTION OF HAUSDORFF MEASURES AND REMOVABILITY 541 

The constant C K depends only on K if h is normalized at infinity requiring h(z) = 

z + O(1/z). For the area, the corresponding estimate was shown in [4]. In fact, as we 

see later, a counterpart of (1.5)forthet-dimensional Hausdorff content M t is the only 

missing detail for proving the absolute continuity φ ∗ H t ′ 

≪ H t for general t. Toward 

solving (1.3), we conjecture that actually, 

M t( h(E) ) ≤ C M t (E), 0


mapping f : \ E → C admits a K-quasiregular extension to . In this definition, 

as in the analytic setting, we may replace L ∞ () by BMO() to get a close variant 

of the problem. 

The sharpness of the bounds in equation (1.2) determines the index 2/(K + 1) 

as the critical dimension in both the L ∞ and the BMO quasiregular removability 

problems. In fact, Iwaniec and Martin [15] previously conjectured that in R n , n ≥ 

2, sets with Hausdorff measure H n/(K+1) (E) = 0 are removable for bounded K- 

quasiregular mappings. A preliminary positive answer for n = 2 was described in [6]. 

Generalizing this, in the present article, we show that, surprisingly, for K>1, one 

can do even better. We have the following improved Painlevé removability. 

THEOREM 1.2 

Let K>1, and suppose that E is any compact set with 

H 2/(K+1) (E) σ -finite. 

Then E is removable for all bounded K-quasiregular mappings. 

The theorem fails for K = 1 since, for instance, the line segment E = [0, 1] is not 

removable for bounded analytic functions. 

For the converse direction, the article [4] finds for every t>2/(K + 1) non-Kremovable 

sets with dim(E) = t. We also make an improvement here and construct 

compact sets with dimension precisely equal to 2/(K + 1) yet not removable for some 

bounded K-quasiregular mappings (for details, see Theorem 5.1). 

Theorems 1.1 and 1.2 are closely connected via the classical Stoïlow factorization, 

which tells (see [6], [18]) that in planar domains, K-quasiregular mappings are 

precisely the maps f representable in the form f = h ◦ φ, where h is analytic and φ 

is K-quasiconformal. Indeed, the first step in proving Theorem 1.2 is to show that for 

a general K-quasiconformal mapping φ, one has 

H 2/(K+1) (E) σ -finite ⇒ H 1( φ(E) ) σ -finite. 

However, this conclusion is not enough since there are rectifiable sets of finite length 

(such as E = [0, 1]) which are nonremovable for bounded analytic functions. Therefore, 

in addition, we need to establish that such “good” sets of positive analytic capacity 

actually behave better also under quasiconformal mappings. That is, we show that up 

to a set of zero length, 

F 1-rectifiable ⇒ dim ( φ −1 (F ) ) > 2 

K + 1 

(for details and a precise formulation, see Corollary 3.2).


The article is structured as follows. In Section 2, we deal with the quasiconformal 

distortion of Hausdorff measures and of other set functions. In Section 3, we study the 

quasiconformal distortion of 1-rectifiable sets. Section 4 gives the proof for the improved 

Painlevé removability theorem for K-quasiregular mappings and other related 

questions. Finally, in Section 5, we describe a construction of nonremovable sets. 

2. Absolute continuity 

There are several natural ways to normalize the quasiconformal mappings φ : C → C. 

In this article, we mostly use the principal K-quasiconformal mappings (i.e., mappings 

that are conformal outside a compact set and are normalized by φ(z) − z = O(1/|z|) 

as |z| →∞). 

It is shown in Astala’s article [4] that for all K-quasiconformal mappings φ : 

C → C, 

|φ(E)| ≤C |E| 1/K , (2.1) 

where C is a constant that depends on the normalizations. By scaling, we may always 

arrange 

diam ( φ(E) ) = diam(E) ≤ 1, (2.2) 

and then C = C(K) depends only on K. In order to achieve the result (2.1), one 

first reduces to the case where the set E is a finite union of disks. Second, applying 

Stoïlow factorization methods, the mapping φ is written as φ = h ◦ φ 1 , where both 

h, φ 1 : C → C are K-quasiconformal mappings, so that φ 1 is conformal on E and 

h is conformal in the complement of the set F = φ 1 (E). Here, one obtains the right 

conclusion for φ 1 , 

|φ 1 (E)| ≤C |E| 1/K , 

by including φ 1 in a holomorphic family of quasiconformal mappings. Further, one 

shows in [4, page 50] that under the special assumption where h is conformal outside 

of F ,wehave 

|h(F )| ≤C |F |, (2.3) 

where the constant C still depends only on K. 

In searching for absolute continuity properties of other Hausdorff measures under 

quasiconformal mappings, such a decomposition seems unavoidable, and this leads 

one to look for counterparts of (2.3) for Hausdorff measures H t or Hausdorff contents 

M t . Here, we establish the result for the dimension t = 1.


LEMMA 2.1 

Suppose that E ⊂ C is a compact set, and let φ : C → C be a principal K- 

quasiconformal mapping, such that φ is conformal on C \ E. Then 

with constants depending only on K. 

M 1( φ(E) ) ≃ M 1 (E) 

In order to prove this result, some background is needed. The space of functions 

of bounded mean oscillation, BMO, is invariant under quasiconformal changes of 

variables (see [26]). More precisely, if φ is a K-quasiconformal mapping and f ∈ 

BMO(C), thenf ◦ φ ∈ BMO(C) with BMO-norm 

‖f ◦ φ‖ ∗ ≤ C(K) ‖f ‖ ∗ . 

The space BMO(C) gives rise to a capacity, 

γ 0 (F ) = sup|f ′ (∞)|, 

where the supremum runs over all functions f ∈ BMO(C) with ‖f ‖ ∗ ≤ 1 

which are holomorphic on C \ F and satisfy f (∞) = 0. Here, f ′ (∞) = 

lim |z|→∞ z (f (z) − f (∞)). Observe that in this situation, ∂f defines a distribution 

supported on F , and actually, |〈∂f,1〉| = |f ′ (∞)|. It turns out (see [32]) that for any 

compact set E, wehave 

γ 0 (E) ≃ M 1 (E). (2.4) 

According to result of Král [17], in the class of functions f ∈ BMO(C) holomorphic 

on C \ E every f admits a holomorphic extension to the whole plane if and only if 

M 1 (E) = 0 (i.e., γ 0 characterizes those compact sets that are removable for BMO 

holomorphic functions). Because of these equivalences, to prove Lemma 2.1, it suffices 

to show that γ 0 (φ(E)) ≃ γ 0 (E). 

Proof 

Suppose that f ∈ BMO(C) is a holomorphic mapping of C \ E such that ‖f ‖ ∗ ≤ 1 

and f (∞) = 0. Then the function g = f ◦ φ −1 is in BMO(C), and‖g‖ ∗ ≤ C(K). 

On the other hand, g is holomorphic on C \ φ(E), and since φ is a principal K- 

quasiconformal mapping, g(∞) = 0,and 

|g ′ (∞)| = lim |zg(z)| = lim |φ(w) f (w)| =|f ′ (∞)|. 

|z|→∞ |w|→∞ 

Hence γ 0 (E) ≤ C(K) γ 0 (φ(E)). The converse inequality follows by symmetry since 

the inverse φ −1 is also a principal mapping.


Lemma 2.1 is a first step toward the results on absolute continuity, as presented in the 

following reformulation of Theorem 1.1. 

THEOREM 2.2 

Let E be a compact set, and let φ : C → C be K-quasiconformal, normalized by 

(2.2). Then 

M 1( φ(E) ) ≤ C ( M 2/(K+1) (E) ) (K+1)/(2K) 

, 

where the constant C = C(K) depends only on K. In particular, if H 2/(K+1) (E) = 0, 

then H 1( φ(E) ) = 0. 

Proof 

There is no restriction if we assume that E ⊂ D. We can also assume that φ is a 

principal K-quasiconformal mapping, conformal outside D.Now,sinceE is compact, 

for any ε>0 there is a finite covering of E by open disks D j , j = 1,...,m, such 

that 

n∑ 

j=1 

r 2/(K+1) 

j ≤ M 2/(K+1) (E) + ε. 

By Vitali’s covering lemma, we can replace our covering by a new finite family of 

disjoint disks, also denoted D j = D(z j ,r j ), j = 1,...,m, such that E is contained 

in the union of 5D j = D(z j , 5r j ). Denote now = ⋃ n 

j=1 5D j.Asin[4], we use 

a decomposition φ = h ◦ φ 1 , where both φ 1 and h are principal K-quasiconformal 

mappings. Moreover, we may require that φ 1 be conformal in ∪ (C \ D) and that h 

be conformal outside φ 1 (). 

By Lemma 2.1, we see that 

M 1( φ(E) ) ≤ M 1( φ() ) = M 1( h ◦ φ 1 () ) ≤ C M 1( φ 1 () ) . 

Hence the problem has been reduced to estimating M 1 (φ 1 ()). For this, K-quasidisks 

have area comparable to the square of the diameter, 

diam ( φ 1 (5D j ) ) ≃ diam ( φ 1 (D j ) ) ( ∫ 1/2 

≃|φ 1 (D j )| 1/2 = J (z, φ 1 ) dA(z)) 

D j 

with constants that depend only on K. Thus, using Hölder estimates twice, we obtain 

n∑ 

diam ( φ 1 (5D j ) ) n∑ 

≃ diam ( φ 1 (D j ) ) ( n∑ 

∫ 

≤ C(K) J (z, φ 1 ) p dA(z) 

D j 

j=1 

j=1 

j=1 

( n∑ 

) 1−1/(2p), 

× |D j | (p−1)/(2p−1) 

j=1 

) 1/(2p)


as long as J (z, φ 1 ) p is integrable. But here we are in the special situation of [7, Lemma 

5.2]. Namely, as φ 1 is conformal in the subset , we may take p = K/(K − 1) and 

apply [7] to obtain 

n∑ 

∫ 

∫ 

J (z, φ 1 ) p dA(z) ≤ 

D j 

j=1 

 

J (z, φ 1 ) p dA(z) ≤ π. 

With the above choice of p, one has (p − 1)/(2p − 1) = 1/(K + 1). Hence we get 

n∑ 

diam ( φ 1 (5D j ) ) ( n∑ ) (K+1)/(2K) 

≤ C(K) r 2/(K+1) 

j 

j=1 

j=1 

≤ C(K) ( M 2/(K+1) (E) + ε ) (K+1)/(2K) 

. (2.5) 

But ⋃ j φ 1(5D j ) is a covering of φ 1 (), so that actually, we have 

M 1( φ(E) ) ≤ CM 1( φ 1 () ) ≤ C(K) ( M 2/(K+1) (E) + ε ) (K+1)/(2K) 

. 

Since this holds for every ε>0, the result follows. 

 

At this point, we emphasize that for a general quasiconformal mapping φ, wehave 

J (z, φ) ∈ L p loc only for p < K/(K − 1). The improved borderline integrability 

(p = K/(K − 1)) under the extra assumption that φ | is conformal was shown in 

[7, Lemma 5.2]. This phenomenon is crucial for our argument since we are studying 

Hausdorff measures rather than dimension. Actually, the same procedure shows that 

inequality (2.5) works in a much more general setting. That is, still under the special 

assumption that φ 1 is conformal in ⋃ n 

j=1 D j,wehaveforanyt ∈ [0, 2], 

( n∑ 

diam ( φ 1 (D j ) ) ) d 

1/d ( n∑ ) (1/t)(1/K), 

≤ C(K) diam(D j ) t (2.6) 

j=1 

where d = 2Kt/(2 + (K − 1)t). On the other hand, another key point in our proof is 

the estimate 

j=1 

M 1( h(E) ) ≤ C M 1 (E), 

valid whenever h is a principal K-quasiconformal mapping that is conformal outside 

E. We believe that finding the counterpart to this estimate is crucial for 

understanding distortion of Hausdorff measures under quasiconformal mappings. We 

make the following conjecture.


CONJECTURE 2.3 

Suppose that we are given a real number d ∈ (0, 2]. Then for any compact set E ⊂ C 

and for any principal K-quasiconformal mapping h which is conformal on C \ E,we 

have 

with constants that depend only on K and d. 

M d( h(E) ) ≃ M d (E) (2.7) 

One may also formulate a convenient discrete variant, which is actually stronger than 

Conjecture 2.3. 

Question 2.4 

Suppose that we are given a real number d ∈ (0, 2] and a finite number of disjoint 

disks D 1 ,...,D n . If a mapping h is conformal on C \ ⋃ n 

j=1 D j and admits a K- 

quasiconformal extension to C, isitthentruethat 

n∑ 

diam ( h(D j ) ) n∑ 

d 

≃ diam(D j ) d (2.8) 

j=1 

with constants that depend only on K and d? 

We already know that (2.7)istrueford = 1 and d = 2; however, for Question 2.4,we 

know a proof only at d = 2. An affirmative answer to Conjecture 2.3, combined with 

the optimal integrability bound proving (2.6), would provide the absolute continuity of 

φ ∗ H d with respect to H t , where d = 2Kt/(2 + (K − 1)t), 0 ≤ t ≤ 2, andK ≥ 1. 

Therefore (2.7) would have important consequences in the theory of quasiconformal 

mappings. 

The positive answer to (2.7) for the dimension d = 1 was based on the equivalence 

(2.4) and the invariance of BMO. Actually, more is true. The space VMO, equal to 

the BMO-closure of uniformly continuous functions, is quasiconformally invariant 

as well. We may also describe VMO, vanishing mean oscillation, as consisting of 

functions f ∈ BMO for which 

lim 1 

|B| 

∫ 

B 

j=1 

|f − f B |=0, 

as |B| +1/|B| → ∞. As we now see, the invariance of VMO has interesting 

consequences.


THEOREM 2.5 

Let E ⊂ C be a compact set, and let φ : C → C be a K-quasiconformal mapping. If 

H 2/(K+1) (E) is finite (or even σ -finite), then H 1 (φ(E)) is σ -finite. 

This result may be equivalently expressed in terms of the lower Hausdorff content. 

To understand this alternative formulation of Theorem 2.5, we first need some background. 

A measure function is a continuous nondecreasing function h(t), t ≥ 0, such 

that lim t→0 h(t) = 0.Ifh is a measure function and F ⊂ C, weset 

M h (F ) = inf ∑ j 

h(δ j ), 

where the infimum is taken over all countable coverings of F by disks of diameter δ j . 

When h(t) = t α , α>0,thenM h (F ) = M α (F ) equals the α-dimensional Hausdorff 

content of F . Moreover, the content M α and the measure H α have the same zero sets. 

We denote by F = F d the class of measure functions h(t) = t d ε(t), 0 ≤ ε(t) ≤ 1, 

such that lim t→0 ε(t) = 0. Thelowerd-dimensional Hausdorff content of F is then 

defined by 

M d ∗ 

(F ) = sup M h (F ). 

h∈F d 

One has M d ∗ ≤ Md , but it can happen that M d ∗ (F ) = 0 < Md (F ). For instance, if F 

is the segment [0, 1] in the plane, then M 1 ∗ (F ) = 0,butM1 (F ) = 1. An old result of 

Sion and Sjerve [28] in geometric measure theory asserts that M d ∗ 

(F ) = 0 if and only 

if F is a countable union of sets with finite d-dimensional Hausdorff measure. For a 

disk B,forM d ∗ (B) = Md (B), and for open sets U, M d ∗ (U) ≃ Md (U). We may now 

reformulate Theorem 2.5 as follows. 

THEOREM 2.6 

Let E ⊂ C be a compact set, and let φ : C → C be a principal K-quasiconformal 

mapping. If M 2/(K+1) 

∗ 

(E) = 0, thenM 1 ∗ 

(φ(E)) = 0. 

For the proof, for any bounded set F ⊂ C define first 

γ ∗ (F ) = sup|f ′ (∞)|, (2.9) 

where the supremum is taken over all functions f ∈ VMO, with ‖f ‖ ∗ ≤ 1, which 

are holomorphic on C \ F and satisfy f (∞) = 0. Again, here we may replace 

|f ′ (∞)| with |〈∂f,1〉|. TheVMO-invariance leads to the following analogue of 

Lemma 2.1.


LEMMA 2.7 

Let E be a compact set. For any principal K-quasiconformal mapping φ : C → C, 

conformal on C \ E, we have 

γ ∗ 

( 

φ(E) 

) 

≃ γ∗ (E). 

Proof 

Consider f ∈ VMOwhich is analytic in C\φ(E) and f (∞) = 0.Setg = f ◦φ.Then 

g ∈ VMO, g is analytic on C \ E, ‖g‖ ∗ ≤ C ‖f ‖ ∗ ,and|g ′ (∞)| =|f ′ (∞)| since φ 

is a principal K-quasiconformal mapping. Consequently, γ ∗ (φ(E)) ≤ Cγ ∗ (E). 

It was shown by Verdera [32] that this VMO-capacity is essentially the 1-dimensional 

lower content. 

LEMMA 2.8 ([32, page 288]) 

For any compact set E, M 1 ∗ (E) ≃ γ ∗(E). 

With these tools, we are ready to prove Theorem 2.6. 

Proof of Theorem 2.6 

Naturally, the argument is similar to that in Theorem 2.2. Without loss of generality, 

we may assume that E ⊂ D and that φ is a principal K-quasiconformal mapping. 

Furthermore, we may assume that H 2/(K+1) (E) is finite, and for any δ, we have a finite 

family of disks D i such that E ⊂ ⋃ i D i, ∑ i diam(D i) 2/(K+1) ≤ H 2/(K+1) (E) + 1, 

and diam(D i )


Since K-quasiconformal mappings are Hölder continuous with exponent 1/K, 

M h( φ 1 () ) ≤ ∑ diam ( φ 1 (D j ) ) ε ( diam(φ 1 (D j )) ) ≤ ε(C K δ 1/K ) ∑ diam ( φ 1 (D j ) ) 

j 

j 

≤ ε(C K δ 1/K ) ∑ j 

( ∫ ) (K−1)/(2K)|Dj 

J (z, φ 1 ) K/(K−1) dm(z) 

| 1/2K 

D j 

( ∑ 

) (K+1)/(2K) 

≤ ε(C K δ 1/K ) C K diam(D j ) 2/(K+1) 

j 

≤ ε(C K δ 1/K ) C K 

(H 2/(K+1) (E) + 1) (K+1)/(2K). 

Finally, taking δ → 0, wegetM h (φ(E)) = 0. This holds for any h ∈ F, andthe 

theorem follows. 

 

One may think of extending the preceding results from the critical index 2/(K + 1) 

to arbitrary ones by using other capacities that behave like a Hausdorff content. For 

instance, the capacity γ α , associated to analytic functions contained in Lip(α) (see 

[23]), satisfies 

M 1+α (E) ≃ γ α (E), 

but unfortunately, the space Lip(α) is not invariant under a quasiconformal change 

of variables. Thus other procedures are needed. It turns out that the homogeneous 

Sobolev spaces provide suitable tools, basically, since Ẇ 1,2 (C) is invariant under quasiconformal 

mappings. Here, recall that for 0


so that g ∈ Ẇ 1,2 (C). In other words, every K-quasiconformal mapping φ induces a 

bounded linear operator 

T : Ẇ 1,2 (C) → Ẇ 1,2 (C), 

T(f ) = f ◦ φ 

with norm depending only on K. As we have mentioned before, this operator T is also 

bounded on BMO(C) (see [26]). Moreover, Reimann and Rychener [27, page 103] 

proved that Ẇ 2/q,q (C), q>2, may be represented as a complex interpolation space 

between BMO(C) and Ẇ 1,2 (C). It follows that T is bounded on the Sobolev spaces 

Ẇ 2/q,q (C),q >2. More precisely, there exists a constant C = C(K,q) such that 

‖f ◦ φ‖ Ẇ 2/q,q (C) ≤ C‖f ‖ Ẇ 2/q,q (C) (2.10) 

for any K-quasiconformal mapping φ on C. These invariant function spaces provide 

us with related invariant capacities. Recall (e.g., see [1, pages 34, 46]) that for any 

pair α>0, p>1 with 0


and consequently, we can write 

f = 1 z ∗ µ = R(I 1 ∗ µ) = I 1−α ∗ R(I α ∗ µ), 

where R is a Calderón-Zygmund operator and ‖f ‖ Ẇ 1−α,q =‖R(I α ∗µ)‖ q ‖I α ∗µ‖ q . 

For the converse, let f = I 1−α ∗ g be an admissible function for γ 1−α,q .Wehave 

that, up to a multiplicative constant, T = ∂f is an admissible distribution for Ċ α,p 

because 

I α ∗ T = R t (g), 

where R t is the transpose of R. Thus Ċ α,p (E) 1/p ≥|〈T,1〉| = |f ′ (∞)|, and the proof 

is complete. 

 

We end up with new quasiconformal invariants built on the Riesz capacities. 

THEOREM 2.10 

Let φ : C → C be a principal K-quasiconformal mapping of the plane which is 

conformal on C \ E.Let1


are compact sets F such that Ċ α,p (F ) = 0 and H h (F ) > 0 for some measure function 

h(t) = t p ε(t). Thus Theorem 2.10 does not help with Conjecture 2.3. Wehavetobe 

content with the following setup. 

Given 1


3. Distortion of rectifiable sets 

In general, if φ is a K-quasiconformal mapping and E is a compact set, it follows 

from (1.2) that 

dim(E) = 1 ⇒ 2 ≤ dim φ(E) ≤ 

2K 

K + 1 

K + 1 . (3.1) 

Here, for both estimates, one may find mappings φ and sets E such that the equality 

is attained (see [4]). In [4], all examples come from nonregular Cantor-type constructions. 

Thus the extremal distortion of Hausdorff dimension is attained, at least, by sets 

irregular enough. The main purpose of this section is to prove that some irregularity 

is also necessary. Namely, we show that quasiconformal images of 1-rectifiable sets 

cannot achieve the maximal distortion of dimension. 

THEOREM 3.1 

Suppose that φ : C → C is a K-quasiconformal mapping, K>1.LetE ⊂ ∂D be a 

subset of the unit circle with dim(E) = 1. Then we have the strict inequality 

dim ( φ(E) ) > 2 

K + 1 . 

With a similar but easier argument, one may also prove that for such sets E, neither 

can dim(φ(E)) attain the upper bound in (3.1) (for details, see Remark 3.7). 

From Theorem 3.1, we obtain as an immediate corollary the following more 

general result. 

COROLLARY 3.2 

Suppose that E is a 1-rectifiable set, and let φ : C → C be a K-quasiconformal 

mapping, K>1.Then 

dim φ(E) > 2 

K + 1 . 

Recall that a set E ⊂ C is said to be 1-rectifiable if there exists a set E 0 of zero length 

such that E \ E 0 is contained in a countable union of Lipschitz curves; that is, 

E \ E 0 ⊂ 

∞⋃ 

j ([0, 1]), 

j=1 

where all j : [0, 1] → C are Lipschitz mappings. Alternatively (see [20]), 1- 

rectifiable sets can be viewed as subsets of countable unions of C 1 -curves, modulo a 

set of zero length. In particular, for any ε>0, there is a decomposition 

∞ 

E \ E ′ 0 = ⋃ 

E i , 

i=1


where E ′ 0 has zero length and each E i can be written as E i = f i (F i ) with f i : C → C 

a (1 + ε)-bi-Lipschitz mapping and F i ⊂ ∂D. From this and Theorem 3.1, we obtain 

Corollary 3.2. 

To prove Theorem 3.1, first some reductions may be made. Recall (see [18]) that 

every K-quasiconformal mapping φ can be factored as φ = φ n ◦···◦φ 1 , where each 

φ j is K j -quasiconformal, and K 1 K 2 ···K n = K. In particular, given ε>0, we can 

choose K j ≤ 1 + ε for all j = 1,...,nwhen n is large enough. On the other hand, 

recall that from the distortion of Hausdorff dimension (1.2), we have 

1 

dim φ(E) − 1 ( 

2 ≤ K 1 

dim E − 1 ) 

. (3.2) 

2 

If φ is such that equality in (3.2) holds for E, then every factor φ j above must give 

equality for the set E j = φ j−1 ◦···◦φ 1 (E) and K = K j . In particular, if the mapping 

φ 1 fails to satisfy the equality in (3.2), then so will φ. By combining these facts, we 

deduce that in order to prove Theorem 3.1, we can assume that K = 1 + ε with ε>0 

as small as we wish. 

For mappings with small dilatation, it is possible to achieve quantitative and more 

symmetric local distortion estimates. In particular, Theorem 3.1 follows from the next 

lower bounds for compression of dimension. 

THEOREM 3.3 

Suppose that φ : C → C is (1 + ε)-quasiconformal, and suppose that E ⊂ ∂D.Then 

for all ε>0 small enough, 

dim(E) ≥ 1 − c 0 ε 2 ⇒ dim ( φ(E) ) ≥ 1 − c 1 ε 2 , (3.3) 

where the constants c 0 ,c 1 > 0 are independent of ε. 

Our basic strategy toward this result is to reduce it to the properties of harmonic measure 

and conformal mappings admitting quasiconformal extensions. Indeed, denote 

by µ the Beltrami coefficient of φ,andleth be the principal solution to ∂h = χ D µ∂h. 

Then h is conformal outside the unit disk. Inside D, it has the same dilatation µ as φ 

and hence differs from this by a conformal factor. Consequently, we may find Riemann 

mappings f : D → := φ(D) and g : D → ′ := h(D) so that 

φ(z) = f ◦ g −1 ◦ h(z), z ∈ D. (3.4) 

Moreover, since the (1 + ε)-quasiconformal mapping G = g −1 ◦ h preserves the disk, 

reflecting across the boundary ∂D one may extend G to a (1 + ε)-quasiconformal 

mapping of C. At the same time, this procedure provides both f and g with (1 + ε) 2 - 

quasiconformal extensions to the entire plane C.


As the final reduction, we now find from (3.4) that for Theorems 3.1 and 3.3,itis 

sufficient to prove the following result. 

THEOREM 3.4 

Suppose that f : C → C is a (1 + ε)-quasiconformal mapping of C, conformal in 

the disk D.LetA ⊂ ∂D. There are constants c 0 , c 1 and γ 0 , γ 1 , independent of ε, such 

that for ε ≥ 0 small enough, 

(i) dim(A) ≥ 1 − c 0 ε 2 ⇒ dim(f (A)) ≥ 1 − c 1 ε 2 , and 

(ii) dim(A) ≤ 1 − γ 0 ε 2 ⇒ dim(f (A)) ≤ 1 − γ 1 ε 2 . 

Proof 

The conclusion (i) follows from Makarov’s fundamental estimates for the harmonic 

measure (see [19]; see also [24, page 231]). In his article [19], Makarov proves that 

for any conformal mapping f defined on D, for any Borel subset A ⊂ ∂D ,andfor 

every q>0, we have the lower bound 

dim ( f (A) ) ≥ 

0 

q dim(A) 

β f (−q) + q + 1 − dim(A) . (3.5) 

Here, β f (p) stands for the integral means spectrum. That is, for a given p ∈ R, β f (p) 

is the infimum of all numbers β such that 

∫ 2π 

( 

|f ′ (re it )| p 1 

) 

dt = O , (3.6) 

(1 − r) β 

as r → 1 − . 

Hence we need estimates for β f (p), and here for mappings admitting K- 

quasiconformal extensions, one has qualitatively sharp bounds. Indeed, it can be 

shown (see [24, page 182]) that 

( K − 1 

) 2 

β f (p) ≤ 9 p 

2 

(3.7) 

K + 1 

for any p ∈ R. The constant 9 is not optimal but suffices for our purposes. Choosing 

q = 1 in (3.5) immediately gives the claim (i). 

For general conformal mappings, there is no bound for expansion of dimension 

(i.e., there is no upper bound analogue of (3.5)). Hence the proof of (ii) strongly 

uses the fact that the mappings considered have (1 + ε)-quasiconformal extensions. 

However, here also, this information is easiest to use in the form (3.7). 

We first need to introduce some further notation. The Carleson squares of the unit 

disk are defined as 

Q j,k = { z ∈ D :2 −k ≤ 1 −|z| < 2 −k+1 , 2 −k+1 πj ≤ arg(z) < 2 −k+1 π(j + 1) } .


Given a point z ∈ D \{0}, letQ(z) denote the unique Carleson square that contains 

z. Then it follows from Koebe’s distortion theorem and quasisymmetry (see [6], [18]) 

that if D(ξ,r) is a disk centered at ξ ∈ ∂D, wehave 

diam ( f (D(ξ,r)) ) ≃ diam ( f (Q(z)) ) ≃|f ′ (z)|(1 −|z|) for z = (1 − r)ξ, (3.8) 

whenever f : C → C is a K-quasiconformal mapping, conformal in D. 

Furthermore, assume that we are given a family of disjoint disks D i = D(ξ i ,r i ) 

with centers ξ i ∈ ∂D, i ∈ N, on the unit circle. Then, write z i = (1 − r i )ξ i ,andfor 

any pair of real numbers 0


then 

(2 −k ) α/δ ≤ (1 −|z i |) α/δ < (1 −|z i |)|f ′ (z i )|≤2 −m . 

Hence the indices m with Im k nonempty lie on an interval m 0 ≤ m ≤ (α/δ)k. 

From Koebe, we also see that if i ∈ Im k ,then|f ′ (w)| p ∼ 2 p(k−m) for every 

w ∈ Q(z i ) with constants depending only on p. Combining this with (3.7) gives,for 

any τ>0, 

q k m ≤ C 2k((9/4)ε2 p 2 +τ+1−((k−m)/k)p) , 

where C now depends on p and τ. We may take p = (k − m)/(10kε 2 ) and obtain 

q k m ≤ C 2(1+τ)k−(k/(10ε)2 )((k−m)/k) 2 . 

Since diam(f (D i )) is comparable to |f ′ (z i )| (1 −|z i |) ∼ 2 −m for i ∈ I k m , 

∑ 

i∈I b (α,δ) 

diam ( f (D i ) ) δ 

≤ C 

∞ 

∑ 

≤ C 

(α/δ)k 

∑ 

q k m 2−mδ 

k=0 m=m 0 

∞∑ 

k=0 

(α/δ)k 

∑ 

m=m 0 

2 k(1+τ−m(δ/k)−(1/(100ε2 ))((k−m)/k) 2) . (3.10) 

One now needs to ensure that the exponent 1 + τ − m(δ/k) − (1/100ε 2 )((k − m)/k) 2 

is negative. In particular, we want the exponent to attain its maximum at m = (α/δ)k, 

andthisissatisfiedif 

α 

δ ≤ 1 − 1 2 (10ε)2 δ. 

Under the assumptions of the lemma, this is easy to verify. Similarly, one verifies that 

the specific choices of the lemma yield the maximum value 

1 + τ − α − 1 (1 − α ) 2 

< 0 

(10ε) 2 δ 

when τ is chosen small enough. It follows that the sum in (3.10) has a finite upper 

bound depending only on the constants M, N. This proves the lemma. 

 

The dimension bounds required in Theorem 3.4(ii) are now easy to establish. For 

every α>1 − γ 0 ε 2 , we have disjoint families of disks D j = D(z j ,r j ) centered on 

∂D and radius r j ≤ ρ → 0 uniformly small, so that A is covered by 5D j and the 

sums ∑ j diam(D j) α are uniformly bounded. On the image side, for each δ>0, 

∑ 

diamf (5D i ) δ ≃ ∑ 

i 

i 

diamf (D i ) δ = 

∑ 

i∈I g (α,δ) 

diamf (D i ) δ + 

∑ 

i∈I b (α,δ) 

diamf (D i ) δ .


As soon as α


Moreover, for the dimension of quasicircles, Smirnov (unpublished) has obtained the 

upper bound 

dim(Ɣ) ≤ 1 + 

( K − 1 

) 2, 

K + 1 

answering a question in [4]. It is still unknown if this bound is sharp; the best known 

lower bounds so far (see [8]) give curves with dimension 

( K − 1 

) 2. 

1 + 0.69 

K + 1 

The arguments that we have used are related to the generalized Brennan conjecture, 

which says that 

β f (p) ≤ p2 

4 

( K − 1 

) 2 

for |p| ≤2 K + 1 

K + 1 

K − 1 , (3.12) 

whenever f is conformal in D and admits a K-quasiconformal extension to C. This 

connection suggests the following. 

Question 3.8 

Let E ⊂ R be a set with Hausdorff dimension 1, andletφ be a K-quasiconformal 

mapping. Then is it true that 

1 − 

( K − 1 

) 2 ( ) ( K − 1 

) 2? 

≤ dim φ(E) ≤ 1 + (3.13) 

K + 1 

K + 1 

The positive answer for the right-hand-side inequality follows from Smirnov’s unpublished 

work, while the left-hand side is only known up to some multiplicative 

constants. On the other hand, Prause [25] proves the left-hand-side inequality for the 

mappings that preserve the unit circle ∂D. 

4. Improved Painlevé theorems 

A compact set E is said to be removable for bounded analytic functions if, for any 

open set with E ⊂ , every bounded analytic function on \ E has an analytic 

extension to . Equivalently, such sets are described by the condition γ (E) = 0, 

where γ is the analytic capacity 

γ (E) = sup { |f ′ (∞)| : f ∈ H ∞ (C \ E),f(∞) = 0, ‖f ‖ ∞ ≤ 1 } . 

Finding a geometric characterization for the sets of zero analytic capacity was a longstanding 

problem. It was solved by David [12] for sets of finite length and, finally, by 

Tolsa [29] in the general case. The difficulties of dealing with this question motivated 

the study of related problems. In particular, we have the question of determining the


removable sets for BMO analytic functions (i.e., those compact sets E such that every 

BMO-function in the plane, holomorphic on C \ E, admits an entire extension). This 

problem was solved by Král [17], who showed that a set E has this BMO-removability 

property if and only if H 1 (E) = 0. This was also proved independently by Kaufman 

[16]. 

For the original case of bounded functions, the Painlevé condition H 1 (E) = 0 

can be weakened. As is well known, there are sets E with zero analytic capacity 

and positive length (e.g., see [14]). In fact, it is now known that among the compact 

sets E with 0 < H 1 (E) < ∞, precisely the purely unrectifiable ones are the removable 

sets for bounded analytic functions (see [12]). Moreover, if E has positive 

σ -finite length, this characterization still remains true, due to the countable semiadditivity 

of analytic capacity (see [29]). 

The preceding problems can be formulated also in the K-quasiregular setting. 

More precisely, a set E is said to be removable for bounded (resp., BMO) K- 

quasiregular mappings if every K-quasiregular mapping in C \ E which is in L ∞ (C) 

(resp., BMO(C)) admits a K-quasiregular extension to C. For simplicity, we use 

here the term K-removable for sets that are removable for bounded K-quasiregular 

mappings. 

Obviously, when K = 1, in both situations of L ∞ and BMO we recover the 

original analytic problem. Moreover, by means of the Stoïlow factorization, one can 

represent any bounded K-quasiregular function as a composition of a bounded analytic 

function and a K-quasiconformal mapping. The corresponding result also holds true 

for BMO since this space, like L ∞ , is quasiconformally invariant. 

Therefore, when we ask ourselves if a set E is K-removable, we just need to 

analyze how it may be distorted under quasiconformal mappings and then apply 

the known results for the analytic situation. With this basic scheme, it is shown in [4, 

Corollary 1.5] that every set with dimension strictly below 2/(K + 1) is K-removable. 

Indeed, the precise formulas for the distortion of dimension (1.2) ensure that for such 

sets, the K-quasiconformal images have dimension strictly smaller than 1. 

Iwaniec and Martin [15] had earlier conjectured that, more generally, sets of zero 

(2/(K + 1))-dimensional measure are K-removable. A preliminary answer to this 

question was found in [6], and actually, it was that argument that suggested Theorem 

2.2. Using our results from above, we can now prove that sets of zero (2/(K + 1))- 

dimensional measure are even BMO-removable. 

COROLLARY 4.1 

Let E be a compact subset of the plane. Assume that H 2/(K+1) (E) = 0. ThenE is 

removable for all BMO K-quasiregular mappings. 

Proof 

Assume that f ∈ BMO(C) is K-quasiregular on C \ E. Denote by µ the Beltrami 

coefficient of f ,andletφ be the principal solution to ∂φ = µ∂φ.ThenF =


f ◦ φ −1 is holomorphic on C \ φ(E) and F ∈ BMO(C). On the other hand, as we 

showed in Theorem 2.2, H 1 (φ(E)) = 0. Thus φ(E) is a removable set for BMO 

analytic functions. In particular, F admits an entire extension, and f = F ◦ φ extends 

quasiregularly to the whole plane. 

 

We believe that Corollary 4.1 is sharp, in the sense that we expect a positive answer 

to the following. 

Question 4.2 

Does there exist for every K ≥ 1, a compact set E with 0 < H 2/(K+1) (E) < ∞, such 

that E is not removable for some K-quasiregular functions in BMO(C)? 

Here, we observe that by [4, Corollary 1.5], for every t>2/(K + 1) there exists a 

compact set E with dimension t, nonremovable for bounded and hence, in particular, 

nonremovable for BMO K-quasiregular mappings. 

Next, we return back to the problem of removable sets for bounded K-quasiregular 

mappings. Here, Theorem 2.2 proves the conjecture of Iwaniec and Martin, that sets 

with H 2/(K+1) (E) = 0 are K-removable. However, the analytic capacity is somewhat 

smaller than length, and hence with Theorem 2.5 we may go even further. If a set 

has finite or σ -finite (2/(K + 1))-measure, then all K-quasiconformal images of E 

have at most σ -finite length. Such images may still be removable for bounded analytic 

functions if we can make sure that the rectifiable part of these sets has zero length. But 

for this, Theorem 3.1 provides exactly the correct tools. We end up with the following 

improved version of the Painlevé theorem for quasiregular mappings. 

THEOREM 4.3 

Let E be a compact set in the plane, and let K>1. Assume that H 2/(K+1) (E) is 

σ -finite. Then E is removable for all bounded K-quasiregular mappings. 

In particular, for any K-quasiconformal mapping φ, the image φ(E) is purely 

unrectifiable. 

Proof 

Let f : C → C be bounded, and assume that f is K-quasiregular on C \ E. Asin 

Corollary 4.1, we may find the principal quasiconformal homeomorphism φ : C → C, 

so that F = f ◦ φ −1 is analytic in C \ φ(E). If we can extend F holomorphically 

to the whole plane, we are done. Thus we have to show that φ(E) has zero analytic 

capacity. 

By Theorem 2.5, φ(E) has σ -finite length (i.e., φ(E) = ⋃ n F n, where each 

H 1 (F n ) < ∞). A well-known result due to Besicovitch (see, e.g., [20, page 205])


assures us that each set F n can be decomposed as 

F n = R n ∪ U n ∪ B n , 

where R n is a 1-rectifiable set, U n is a purely 1-unrectifiable set, and B n is a set of 

zero length. Because of the semiadditivity of analytic capacity (see [29]), 

γ (F n ) ≤ C ( γ (R n ) + γ (U n ) + γ (B n ) ) . 

Now, γ (B n ) ≤ C H 1 (B n ) = 0 and γ (U n ) = 0 since purely 1-unrectifiable sets 

of finite length have zero analytic capacity (see [12]). On the other hand, R n is 

a 1-rectifiable image, under a K-quasiconformal mapping, of a set of dimension 

2/(K + 1). Thus applying Theorem 3.1 and Corollary 3.2 to φ −1 shows that we must 

have H 1 (R n ) = 0. Therefore we get γ (F n ) = 0 for each n. Again, by countable 

semiadditivity of analytic capacity, we conclude that γ (φ(E)) = 0. 

 

As pointed out earlier, Theorem 4.3 does not hold for K = 1.Any1-rectifiable set such 

as E = [0, 1] of finite and positive length gives a counterexample. In the above proof, 

the improved distortion of 1-rectifiable sets is the decisive phenomenon allowing the 

result. In fact, such “good” behavior of rectifiable sets has further consequences. For instance, 

even strictly above the critical dimension 2/(K + 1) = 1 − (K − 1)/(K + 1), 

one may find removable sets, as soon as they have enough geometric regularity. 

COROLLARY 4.4 

There exists a constant c ≥ 1 such that if E ⊂ ∂D is compact and 

( K − 1 

) 2, 

dim(E) < 1 − c 

K + 1 

then E is removable for bounded and BMO K-quasiregular mappings, K = 1 + ε, 

whenever ε>0 is small enough. 

Proof 

This is a consequence of Corollary 3.6.Ifε>0 is small enough and K = 1 + ε,then 

the K-quasiconformal images of E always have dimension strictly below 1, sothat 

γ (φ(E)) = 0 for each K-quasiconformal mapping φ. 

 

In conjunction with Question 3.8, we have the following. 

Question 4.5 

Let K > 1. Then is every set E ⊂ ∂D with dim(E) < 1 − ((K − 1)/(K + 1)) 2 

removable for bounded and BMO K-quasiregular mappings?


5. Examples of extremal distortion 

Sections 2 and 3 provide a delicate analysis of distortion of 1-dimensional sets under 

quasiconformal mappings but still leave open the cases where dim(E) = 2/(K + 1) 

precisely, but E does not have σ -finite (2/(K + 1))-measure. Hence we are faced with 

the natural question: are there compact sets E, with dim(E) = 2/(K + 1), which are 

not removable for some bounded K-quasiregular mappings? 

In this last section, we give a positive answer and show that our results are sharp 

in quite a strong sense. Indeed, to compare with the analytic removability, recall first 

that by Mattila’s theorem [21, Theorem 3.8], if a compact set E supports a probability 

measure with µ(B(z, r)) ≤ rε(r) and 

∫ 

ε(t) 2 

dt < ∞, (5.1) 

0 t 

then the analytic capacity γ (E) > 0. On the other hand, if the integral in (5.1) diverges, 

then there are compact sets E of vanishing analytic capacity supporting a probability 

measure with µ(B(z, r)) ≤ rε(r) (see [29]). In a complete analogy, we prove the 

following. 

THEOREM 5.1 

Let K ≥ 1. Suppose that h(t) = t 2/(K+1) ε(t) is a measure function such that 

∫ 

ε(t) 1+1/K 

dt < ∞. (5.2) 

0 t 

Then there is a compact set E that is not K-removable and yet supports a probability 

measure µ with µ(B(z, r)) ≤ h(r) for every z and r>0. 

In particular, whenever ε(t) is chosen so that, in addition, for every α>0, we 

have t α /ε(t) → 0 as t → 0, then the construction gives a non-K-removable set E 

with dim(E) = 2/(K + 1). 

Proof 

We construct a compact set E and a K-quasiconformal mapping φ so that H h (E) ≃ 1 

and, at the same time, φ(E) has a positive and finite H h′ -measure for some measure 

function h ′ (t) = tε ′ (t), where 

h ′ (t) = tε ′ (t) 

with 

∫ 1 

0 

ε ′ (t) 2 

dt < ∞. 

t 

Then Mattila’s theorem [21, Theorem 3.8] shows that γ (φ(E)) > 0, so that there exist 

nonconstant bounded functions h holomorphic on C \ φ(E). Thus, with f = h ◦ φ, 

we see that E is not removable for bounded K-quasiregular mappings.


We construct the K-quasiconformal mapping φ as the limit of a sequence φ N 

of K-quasiconformal mappings, and E is a Cantor-type set. To reach the optimal 

estimates, we need to change, at every step in the construction of E, both the size and 

the number m j of the generating disks. 

Without loss of generality, we may assume that for every α>0, t α /ε(t) → 0 as 

t → 0. 

Step 1. First, choose m 1 disjoint disks D(z i ,R 1 ) ⊂ D, i = 1,...,m 1 ,sothat 

c 1 := m 1 R 2 1 ∈ ( 1 

2 , 1 ). 

For R 1 small enough (i.e., for m 1 large enough), this is clearly possible. The function 

f (t) = m 1 h(tR 1 ) is continuous with f (0) = 0. Moreover, for each fixed t, 

f (t) = m 1 (tR 1 ) 2/(K+1) ε(tR 1 ) = 

ε(t √ c 1 /m 1 ) 

(t √ c 1 /m 1 ) 2K/(K+1) t 2 c 1 → ∞ 

as m 1 →∞. Hence, for any t < 1, we may choose m 1 so large that there exists 

σ 1 ∈ (0,t) satisfying m 1 h(σ K 1 R 1) = 1. A simple calculation gives 

m 1 σ 1 R 1 ε(σ K 1 R 1) (K+1)/(2K) (c 1 ) (1−K)/(2K) = 1. (5.3) 

Next, let r 1 = R 1 . For each i = 1,...,m 1 ,letϕ 1 i (z) = z i + σ K 1 R 1 z, and using 

the notation αD(z, ρ) := D(z, αρ),set 

D i := 1 

σ K 1 

ϕ 1 i (D) = D(z i,r 1 ), 

D ′ i := ϕ1 i (D) = D(z i,σ K 1 r 1) ⊂ D i . 

As the first approximation of the mapping, define 

⎧ 

σ ⎪⎨ 

1−K 

1 (z − z i ) + z i , z ∈ D 

i ′ , 

g 1 (z) = ∣ z − z ∣ 

i ∣∣ 

1/K−1 

(z − zi ) + z i , z ∈ D i \ D 

i ′ r , 

1 

⎪⎩ z, z /∈ ∪D i . 

This is a K-quasiconformal mapping, conformal outside of ⋃ m 1 

i=1 (D i \ D 

i ′ ). It maps 

each D i onto itself and D 

i ′ onto D′′ 

i 

= D(z i ,σ 1 r 1 ), while the rest of the plane remains 

fixed. Write φ 1 = g 1 . 

Step 2. We have already fixed m 1 ,R 1 ,σ 1 ,andc 1 . Consider m 2 disjoint disks of radius 

R 2 , centered at z 2 j , j = 1,...,m 2, uniformly distributed inside of D, sothat 

c 2 = m 2 R 2 2 > 1 2 .


Figure 1 

Then repeat the above procedure, and choose m 2 so large that the equation 

m 1 m 2 h(σ K 1 σ K 2 R 1R 2 ) = 1 

has a unique solution σ 2 ∈ (0, 1), as small as we wish. Then 

m 1 m 2 σ 1 σ 2 R 1 R 2 ε(σ K 1 σ K 2 R 1R 2 ) (K+1)/(2K) (c 1 c 2 ) (1−K)/(2K) = 1. 

Denote r 2 = R 2 σ 1 r 1 and ϕ 2 j (z) = z2 j + σ K 2 R 2 z, and define the auxiliary disks 

( 1 

) 

D ij = φ 1 ϕ 1 

σ2 

K i 

◦ ϕ 2 j (D) = D(z ij ,r 2 ), 

D ′ ij = φ ( 

1 ϕ 

1 

i 

◦ ϕ 2 j (D)) = D ′ (z ij ,σ K 2 r 2) 

for certain z ij ∈ D, where i = 1,...,m 1 and j = 1,...,m 2 .Now,let 

⎧ 

σ ⎪⎨ 

1−K 

2 (z − z ij ) + z ij , z ∈ D 

ij ′ , 

g 2 (z) = ∣ z − z ∣ 

ij ∣∣ 

1/K−1 

(z − zij ) + z ij , z ∈ D ij \ D 

ij ′ r , 

2 

⎪⎩ z, otherwise. 

Clearly, g 2 is K-quasiconformal, conformal outside of ⋃ i,j (D ij \D 

ij ′ ), and maps each 

D ij onto itself and D 

ij ′ onto D′′ 

ij = D(z ij ,σ 2 r 2 ), while the rest of the plane remains 

fixed. Define φ 2 = g 2 ◦ φ 1 . 

The induction step (see Figure 1). After step N − 1, wetakem N disjoint disks of 

radius R N , with union of D(z N l ,R N ) covering at least half of the area of D, 

c N = m N R 2 N > 1 2 . (5.4)


As before, we may choose m N so large that m 1 ···m N h(σ K 1 ···σ K N R 1 ···R N ) = 1 

holds for a unique σ N , as small as we wish. Note that lim N→∞ σ N = 0 and 

m 1 ···m N σ 1 R 1 ···σ N R N ε(σ K 1 R 1 ···σ K N R N) (K+1)/(2K) (c 1 ···c N ) (1−K)/(2K) = 1. 

Then, denote ϕj N(z) = zN j + σN KR N z and r N = R N σ N−1 r N−1 . For any multi-index 

J = (j 1 ,...,j N ), where 1 ≤ j k ≤ m k , k = 1,...,N,let 

D J = φ N−1 

( 1 

σ K N 

) 

ϕ 1 j 1 

◦···◦ϕ N j N 

(D) = D(z J ,r N ), 

D ′ J = φ ( 

N−1 ϕ 

1 

j 1 

◦···◦ϕ N j N 

(D) ) = D ′ (z J ,σ K N r N), 

and let 

⎧ 

σ ⎪⎨ 

1−K 

N (z − z J ) + z J , z ∈ D 

J ′ , 

g N (z) = ∣ z − z ∣ 

J ∣∣ 

1/K−1 

(z − zJ ) + z J , z ∈ D J \ D 

J ′ r , 

N 

⎪⎩ z, otherwise. 

Clearly, g N is K-quasiconformal, conformal outside of ⋃ J =(j 1 ,...,j N ) (D J \ D 

J ′ ),and 

maps D J onto itself and D 

J ′ onto D′′ 

J 

= D(z J ,σ N r N ), while the rest of the plane 

remains fixed. Now, define φ N = g N ◦ φ N−1 . 

Since each φ N is K-quasiconformal and equals the identity outside the unit disk 

D, there exists a limit K-quasiconformal mapping 

φ = lim 

N→∞ φ N 

with convergence in W 1,p 

loc (C) for any p


Observe that we have chosen the parameters R N ,m N ,σ N so that 

m 1 ···m N h(s N ) = 1, (5.6) 

m 1 ···m N t N ε(s N ) (K+1)/(2K) (c 1 ···c N ) (1−K)/(2K) = 1. (5.7) 

Claim. We have H h (E) ≃ 1. 

Since diam(ϕ 1 j 1 

◦···◦ϕ N j N 

(D)) ≤ δ N → 0 when N →∞,wehave,by(5.6), 

∑ 

H h (E) = lim H h δ 

(E) ≤ lim h ( diam(ϕ 1 j δ→0 δ→0 

1 

◦···◦ϕ N j N 

(D)) ) = m 1 ···m N h(s N ) = 1. 

j 1 ,...,j N 

For the converse inequality, take a finite covering (U j ) of E by open disks of diameter 

diam(U j ) ≤ δ, andletδ 0 = inf j (diam(U j )) > 0. Denote by N 0 the minimal integer 

such that s N0 ≤ δ 0 . By construction, the family (ϕ N 0 

j N0 

◦···◦ϕj 1 1 

(D)) j1 ,...,j N0 

is a covering 

of E with the M h -packing condition (see [20]). Thus 

∑ 

h ( diam(U j ) ) ≥ C 

j 

∑ 

Hence H h δ (E) ≥ C, and letting δ → 0, weget 

j 1 ,...,j N0 

h ( diam(ϕ N 0 

j N0 

◦···◦ϕ 1 j 1 

(D)) ) = C. 

C ≤ H h( φ(E) ) ≤ 1, 

proving our first claim. 

A similar argument, based this time on (5.7), gives that H h′ (φ(E)) ≃ 1 for a 

measure function h ′ (t) = tε ′ (t), as soon as for all indices N, 

ε ′ (t N ) = ε(s N ) (K+1)/(2K) (c 1 ···c N ) (1−K)/(2K) . (5.8) 

Claim. One can find a continuous and nondecreasing function ε ′ (t) satisfying (5.8) 

and 

∫ 1 

ε ′ (t) 2 

dt < ∞. (5.9) 

0 t 

Indeed, let us first choose a continuous nondecreasing function v(t) so that v(t) → 0 

as t → 0 and so that (5.2) still holds in the form 

∫ 

ε(t) 1+1/K 

dt < ∞. (5.10) 

tv(t) 

0


In the above inductive construction, we can then choose the σ j ’s so that 

v(σ1 

K ···σ N K) ≤ 2−N(1−1/K) for every index N. Now,(5.4)and(5.8) imply that 

ε ′ (t N ) 2 ≤ ε(s N ) 1+1/K 2 N(1−1/K) ≤ ε(s N) 1+1/K 

. 

v(s N ) 

On the other hand, by (5.5), we also have t N−1 /t N ≤ s N−1 /s N , and so we may extend 

ε ′ (t), determined by (5.8) only at the t N ’s, so that it is continuous, nondecreasing, and 

satisfies 

∫ 

0 

ε ′ (t) 2 dt 

t 

∫ 

≤ 

0 

ε(s) 1+1/K 

v(s) 

ds 

s < ∞. 

Hence the claim follows. Combining it with Mattila’s theorem [21, Theorem 3.8] 

completes the proof of the theorem. 

 

Lastly, let us note that if we do not care for the analytic capacity of the target set, a 

straightforward modification of Theorem 5.1, normalizing the disks of the construction 

so that m N t N η(t N ) = 1, gives the following. 

COROLLARY 5.2 

Let K ≥ 1, and let h(t) = tη(t) be a measure function such that 

• η is continuous and nondecreasing, η(0) = 0, and η(t) = 1 whenever t ≥ 1; 

• lim 

t→0 

(t α /η(t)) = 0 for all α>0. 

There exist a compact set E ⊂ D and a K-quasiconformal mapping φ such that 

dim(E) = 2 

K + 1 

and H h( φ(E) ) = 1. (5.11) 

Note added in proof. In a recent work, Bishop [10] has given a negative answer to 

Question 2.4. However, Conjecture 2.3 remains open. 

On the other hand, Uriarte-Tuero [31] has recently given a positive answer to 

Question 4.2. 

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549 

Astala 

Department of Mathematics and Statistics, University of Helsinki, FI-00014 Helsinki, Finland; 

kari.astala@helsinki.fi 

Clop 

Departament de Matemàtiques, Facultat de Ciències, Universitat Autònoma de Barcelona, 

08193 Bellaterra, Barcelona, Catalonia; albertcp@mat.uab.cat 

Mateu 


08193 Bellaterra, Barcelona, Catalonia; mateu@mat.uab.cat 

Orobitg 


08193 Bellaterra, Barcelona, Catalonia; orobitg@mat.uab.cat 

Uriarte-Tuero 

Department of Mathematics, University of Missouri–Columbia, Columbia, Missouri 

65211-4100, USA; ignacio@math.missouri.edu

SOME ASYMPTOTICS OF TOPOLOGICAL 

QUANTUM FIELD THEORY VIA SKEIN THEORY 

JULIEN MARCHÉ and MAJID NARIMANNEJAD 

Abstract 

For each oriented surface of genus g, we study a limit of quantum representations of 

the mapping class group arising in topological quantum field theory (TQFT) derived 

from the Kauffman bracket. We determine that these representations converge in the 

Fell topology to the representation of the mapping class group on H(), the space of 

regular functions on the SL(2, C)-representation variety with its Hermitian structure 

coming from the symplectic structure of the SU(2)-representation variety. As a corollary, 

we give a new proof of the asymptotic faithfulness of quantum representations. 


A topological quantum field theory (TQFT) in dimension 2 + 1 is an algebraic structure 

very close to topology: roughly speaking, it associates to each surface a finitedimensional 

vector space and to each cobordism a linear map between the vector 

spaces associated to the boundaries. Such theories have physical origins: they were 

introduced by Witten [W] in the 1980s from Chern-Simons actions and generated 

very rich mathematical developments. There are various rigorous constructions coming 

from quantum groups (see [RT]), geometric quantization, and many other areas. 

Unfortunately, such constructions remain complicated, and it is hard to make concrete 

computations. 

In this article, we prefer the approach of [BH+2], which defines TQFTs in a 

purely combinatorial way: using skein theory and the Kauffman bracket, the authors 

define a family of Hermitian TQFTs (V p , 〈·, ·〉 p ) corresponding for p = 2r to the 

SU(2)-theory at level r − 2. Despite the simple and very beautiful structure of these 

combinatorial TQFTs, the connection with geometry is less clear than from other 

approaches. In this article, we show that some connections can be found in a simple 

and direct way. From the axioms, a TQFT generates for any closed surface a family 

of representations of the extended mapping class group of , a central extension of 

the mapping class group by Z coming from p 1 -structures (see [MR]). In some sense, 

these representations carry the main topological meaning of TQFTs. Hence we link 


Vol. 141, No. 3, c○ 2008 DOI 10.1215/00127094-2007-006 

Received 7 June 2006. Revision received 15 June 2007. 

2000 Mathematics Subject Classification. Primary 57M27; Secondary 57M50, 37E30. 

573

574 MARCHÉ and NARIMANNEJAD 

them with some geometrical representation. The basic idea for this comes from a 

general belief that when p goes to infinity, things become classical, by which we 

mean geometrical. This belief is based on the so-called semiclassical approximation. 

Hence we study the limit of ρ p , the quantum representations of Ɣ g on V p (). 

For this purpose, let us describe two classical spaces on which the mapping class 

group acts. 

Fix a closed oriented surface of genus g. We call multicurve an isotopy class of 

1-dimensional submanifold of without component bounding a disc. (The empty set 

is also considered as a multicurve.) The mapping class group of Ɣ g acts on the set of 

multicurves in a natural way. Call C() the C vector space generated by multicurves; 

we obtain a representation of Ɣ g on C(). This fundamental representation carries 

almost all information about the structure of Ɣ g . For instance, no nontrivial element 

of Ɣ g acts trivially on multicurves, except for the elliptic and hyperelliptic involutions 

in genus 1 and 2. 

Another very natural space on which the mapping class group acts is 

hom(π 1 (),G)/G, theG-character variety of π 1 () for a fixed Lie group G. Let 

us denote it by S(,G). Any element of the mapping class group of may be 

represented as an automorphism of π 1 (); its action on S(,G) is then obtained 

by left composition on hom(π 1 (),G). In that way, we also obtain an action of 

the mapping class group on any space of functions defined on the G-character 

variety. 

We are interested here in the cases where G = SU(2) and G = SL(2, C). 

These spaces have a rich structure; we use the natural symplectic structure ω on 

the smooth part of S(,SU(2)) (see [G1]) and the structure of an algebraic variety 

on S(,SL(2, C)). WedefineH() as the ring of regular functions on S(, 

SL(2, C)). 

Using the natural inclusion of S(,SU(2)) in S(,SL(2, C)), we can define a 

Hermitian form on H() by the formula 

∫ 

〈f, g〉 = f gdV. 

S(,SU(2)) 

Here, dV is the volume form on S(,SU(2)) induced by the symplectic form ω. 

The following theorem can be interpreted in terms of the Fell topology. For 

convenience, let us recall this notion briefly. Let G be a discrete group, and let 

ρ k : G → U(V k ) be a sequence of unitary representations of G into V k . One says that 

this sequence converges to the representation ρ : G → U(V ) in the Fell topology 

if, for any unit vector v ∈ V and any finite subset S ⊂ G, there is a sequence of 

unit vectors v k ∈ V k such that for all g ∈ S, the sequence 〈ρ k (g)v k ,v k 〉 converges to 

〈ρ(g)v, v〉. 

We obtain the following result.

SOME ASYMPTOTICS OF TQFT VIA SKEIN THEORY 575 

THEOREM 

Let be a closed oriented surface of genus g. For all even integers p, thereisa 

Ɣ g -equivariant map ϕ p : H() → End(V p ()) such that 

2 

) d(g)〈ϕp 

〈v, w〉 = lim (v),ϕ 

p→∞( 

p (w)〉 p for all v, w ∈ H(). 

p 

Here, we have set d(1) = 1 and d(g) = 3g − 3 for g>1. The Hermitian form on 

End(V p ()) is defined by 〈x,y〉 p = Tr(xy ∗ ). This implies, in particular, that the quantum 

representations ρ p ⊗ ρ p converge in the Fell topology to ρ : Ɣ g → U(H()), 

the natural representation coming from the action of Ɣ g on S(,SL(2, C)). 

We obtain as a corollary a new proof of the following result of [A1](seealso[FWW]) 

about asymptotic faithfulness of quantum representations. 

COROLLARY 

Let be a closed oriented surface of genus g. For any nontrivial h in Ɣ g which is not 

the elliptic (g = 1) or hyperelliptic (g = 2) involution, there is some even p 0 such 

that ρ p (h) ≠ Id for all even p ≥ p 0 . 

Proof 

One can associate to any curve γ on a regular function f γ on S(,SL(2, C)) 

by the formula f γ (ρ) =−Tr ρ(γ ). For a disjoint union of curves, we associate the 

product of the functions associated to each component. In this way, we construct a 

map f from C() to H(). By a result of [B, Theorem 10] and [PS, Theorem 4.7], 

the map f is an isomorphism of vector spaces. Therefore, we can think of a regular 

function on S(,SL(2, C)) as a linear combination of multicurves. 

Recall that no element of Ɣ g acts trivially on C() except the identity and the 

elliptic and hyperelliptic involutions in genus 1 and genus 2. Hence we can suppose 

that there is some v in C() ≃ H() such that w = hv − v is nonzero. This implies 

that 〈w, w〉 is nonzero because the form 〈·, ·〉 is nondegenerate. 

In fact, if 〈w, w〉 =0, then the regular function on S(,SL(2, C)) associated 

to w satisfies 

∫ 

|w| 2 dV = 0. 

S(,SU(2)) 

As w is continuous, it must vanish on S(,SU(2)). Moreover, as it is holomorphic 

on the space S(,SL(2, C)) and zero on S(,SU(2)), it vanishes identically (see 

[G2, proof of Theorem 1.4.1]). 

Due to the equality 〈w, w〉 =lim p→∞ (2/p) d(g) 〈ϕ p w, ϕ p w〉 p , we can find even 

p 0 such that for all even p ≥ p 0 , ϕ p w ≠ 0. Hence ϕ p (hv) ≠ ϕ p (v),andρ p (h) cannot 

be the identity.


1.1. Proof of the theorem 

The heart of the proof is the construction of the map ϕ p , which is almost obvious but 

is fundamental. As the space H() is isomorphic to C(), to define the map ϕ p it is 

sufficient to construct ϕ p (γ ) ∈ V p () ⊗ V p () ∗ for any multicurve γ . 

For such a multicurve, we consider the cobordism × [0, 1] with the multicurve 

embedded as γ ×{1/2}. The TQFT naturally induces an element Z p ( × [0, 1],γ) 

in V p (∐ − ) = V p () ⊗ V p () ∗ . We call this element ϕ p (γ ). We remark that 

it defines a self-adjoint element of End(V p ()) as the pair made of the cobordism 

× [0, 1] and the curve γ inside is isomorphic to the same pair with opposite 

orientations and exchanged boundaries. This gives our fundamental map ϕ p ,whichis 

clearly equivariant because of the naturality of the construction. 

To prove the theorem, one has to compute the limit of the expression 

(2/p) d(g) 〈ϕ p (γ ),ϕ p (δ)〉 p for two multicurves γ and δ. 

We do this in two steps. In the first step, we assume that δ is empty. Using 

combinatorial techniques from [BH+2], we obtain for 〈ϕ p (γ ), 1〉 p an explicit formula 

resembling a Riemann sum. When we normalize it, it converges to an integral over a 

subspace of R d(g) ; we denote its value by 〈γ 〉. By linearity, we extend 〈·〉 to a map 

from C() to C. 

In the second step, we use the connection between the TQFT V p and the Kauffman 

skein module at A =−e iπ/p . We easily find that (2/p) d(g) 〈ϕ p (γ ),ϕ p (δ)〉 p converges 

to 〈γ · δ〉, where · is the multiplication induced on C() by its identification with the 

Kauffman skein algebra of × [0, 1] at A =−1 (see [PS]). 

On the other hand, it is well known that this multiplication on C() is isomorphic 

to the natural multiplication on H(), the space of regular functions on 

S(,SL(2, C)) (see [B], [PS]). 

It remains to identify the linear form on H() defined by f γ ↦→〈γ 〉. Suppose 

that γ is a multicurve. We choose curves C i on which decompose the surface 

into pants such that all components of γ are parallel to some C i .Itiswellknown 

that the maps f i = f Ci form a system of Poisson commuting functions on S(, 

SU(2)). 

As shown in [JW], the product of the maps (f i ):S(,SU(2)) → R d(g) is the 

moment map for an action of a torus of dimension d(g) on a dense open subset of 

S(,SU(2)). The authors use the Duistermaat-Heckman theorem to give an explicit 

formula for the volume form dV on S(,SU(2)). From their result, we deduce the 

following striking formula: 

∫ 

〈γ 〉= 

S(,SU(2)) 

The theorem follows from this formula. 

f γ dV.


1.2. Remarks and perspectives 

The main motivation for this work came from the article [FK], which is about the 

asymptotics of quantum representations of the mapping class group of the torus. Our 

approach is different in the sense that we study the limit of V p ⊗ Vp ∗ instead of simply 

V p . We were also inspired by the ideas contained in the works [F]and[M]. Our work 

is, of course, related to the article [A1], where similar ideas appear, and also has some 

intersection with [BFK]. 

Questions 

There are many questions naturally linked to our results. 

(1) How can we link our asymptotic result to the asymptotics considered in [FK]? 

(2) Can we apply our result or some refinements thereof to the problem of 

Andersen, Masbaum, and Ueno in [AMU, Question 1.1]? The Nielsen- 

Thurston classification of the elements of the mapping class group is directly 

related to their action on multicurves. As the quantum representations converge 

to this action, can we find some trace of this classification in quantum representations 

at finite level? Some ideas with respect to this question are developed 

in [A3]. 

(3) In [BH+2], one can choose any primitive 4r root of unity to construct a TQFT. 

We have chosen roots converging to −1. Is it possible to develop the same 

asymptotics for roots of unity converging to different complex numbers? 

(4) Can we obtain a stronger convergence for the sequence involved in the theorem? 

For instance, what is the expansion of this sequence into powers of 1/p? 

The results of this article were recovered and generalized to the SU(n)-case via the 

theory of Toeplitz operators in a purely geometric framework (see [A2]). 

2. Review of TQFT 

This part is a quick and formal review of the TQFT constructed in [BH+2]whichwe 

give to fix notation and settings and to recall results that are used in this article. We 

refer the interested reader to that beautiful original article. 

Fixanevenintegerp = 2r. The complex number A =−e iπ/(2r) is a primitive 

4rth root of unity. One can construct from it a 2 + 1 TQFT. 

In the notation of [BH+2], we set κ = e −iπ/(2r)−iπ(2r+1)/12 and η = 

( √ 2/r) sin(π/r). WedefineC r ={0, 1,...,r − 2} to be the set of colors. 

Atriple(a,b,c) of elements of C r is called r-admissible if a + b + c is even, 

the triangle inequality |a − b| ≤c ≤ a + b is satisfied, and, moreover, we have 

a + b + c


2.1. The cobordism category 

A TQFT is a linear representation of a cobordism category. In our settings, the objects 

of our category are oriented surfaces with marked points and p 1 -structures. 

• A marking of a surface is a finite family (z j ,c j ) j∈J , where (z j ) is a family 

of distinct points in with, for all j ∈ J , a nonzero tangent direction v j at z j 

on . Forallj ∈ J , c j is a color in C r . 

• A p 1 -structure is a somewhat complicated object used to solve the so-called 

framing anomaly. Consider the map p 1 : BO → K(Z, 4) corresponding to 

the first Pontryagin class. Let X be its homotopy fiber, that is, the set of couples 

(x,γ) ∈ BO × C([0, 1],K(Z, 4)) satisfying γ (0) =∗,andγ (1) = p 1 (x). 

Let E be the universal stable bundle over BO,andletE X be its pullback over 

X. Ap 1 -structure on a manifold M is a fiber map from the stable tangent 

bundle of M to E X . 

In the notation of an object (,z,c), we do not mention the directions v j and the 

p 1 -structure, although they are present. 

Now, we define morphisms. Let ( 1 ,z 1 ,c 1 ) and ( 2 ,z 2 ,c 2 ) be two objects as 

defined above. A morphism is 

• an oriented 3-manifold M whose boundary is decomposed as ∂M =− 1 ∐ 2 , 

where − means with opposite orientation; 

• a colored, banded trivalent graph G embedded in M whose restriction to the 

boundary is compatible with the marked points; 

• a p 1 -structure on M extending the p 1 -structure given on the boundary. 

A banded trivalent graph G in M is a graph with monovalent or trivalent vertices 

contained in an oriented surface SG ⊂ M such that 

(i) G meets ∂M transversally on the set of 1-valent vertices of G noted ∂G; 

(ii) the surface SG is a regular neighborhood of G in SG, andSG ∩ ∂M is a 

regular neighborhood of G ∩ ∂M in SG ∩ ∂M. 

A coloring of G is a map σ from the set of edges of G to C r such that the colors of 

the edges meeting at each vertex are r-admissible. The restriction of a banded graph 

G ⊂ M on ∂M gives marked points (z j ) j∈J with tangent directions (v j ) j∈J , whereas 

the restriction of a coloring gives colors (c j ) j∈J . 

Two morphisms are called equivalent if the corresponding manifolds are isomorphic, 

the banded graphs are isotopic, and the p 1 -structures are homotopic relative to 

the boundary. 

2.2. Main properties of TQFT 

Theorem 1.4 in [BH+2] states that for each integer p, there is a functor (V p ,Z p ) 

from the precedent cobordism category to the category of finite-dimensional C vector 

spaces.


This means that to every object (,z,c), we can associate a vector space 

V p (,z,c), and to any morphism (M,G) between two objects, we can associate 

a linear map Z p (M,G) between the vector spaces corresponding to the objects. By 

convention, V p (∅) = C; hence any closed manifold (M,G) acts as a scalar 〈M,G〉 p 

that is a 3-manifold invariant. Moreover, there is natural Hermitian form 〈·, ·〉 p on 

V p (,z,c) such that for any two morphisms (M 1 ,G 1 ) and (M 2 ,G 2 ) from ∅ to 

(,z,c),wehave〈Z p (M 1 ,G 1 ),Z p (M 2 ,G 2 )〉 p =〈M 1 ∪ (−M 2 ),G 1 ∪ G 2 〉 p . 

We give here some important results related to this construction. 

THEOREM 2.1 ([BH+1, Theorem 4.11]) 

Let (,z,c) be a surface with marked points and p 1 -structure. Let H be a handlebody 

whose boundary is and with a p 1 -structure extending that of . LetG be a 

banded graph with monovalent or trivalent vertices in H such that monovalent vertices 

correspond to marked points z and such that H is a tubular neighborhood of G. For 

each coloring σ of G compatible with the coloring of the boundary, we denote by u σ 

the element induced by Z p in V p (,z,c). 

Then the elements u σ form an orthogonal basis of V p (,z,c), and if G does not 

contain any closed loop, we have 

∏ 

〈u σ ,u σ 〉 p = η #v−#e ∏ v〈σ v〉 

e 〈σ e〉 . 

In this formula, v ranges over the set of vertices of G, and e ranges over the set of 

edges. Moreover, for any trivalent vertex v, σ v is the triple of colors of the edges 

adjacent to this vertex, and for any monovalent vertex v, σ v is the color of the edge 

incoming to it. 

We set 〈j〉 =(−1) j [j + 1] and 〈a,b,c〉 =(−1) α+β+γ ([α + β + γ + 1]![α]! 

[β]![γ ]!/([a]![b]![c]!)), whereα, β, and γ are defined by the equations a = β + 

γ,b = α + γ, and c = α + β. 

If G is reduced to a closed loop, then the formula is simply 〈u σ ,u σ 〉 p = 1. 

Remark 2.2 

We check that for our choice of root of unity A and for a surface without marked 

points, the Hermitian pairing on V p () is positive definite. 

2.3. Kauffman bracket and TQFT 

We define K(M) as the usual skein module of any oriented 3-manifold M. We refer 

to [PS] for a complete account, but we recall here what we need. Let A be some 

indeterminate. The Z[A, A −1 ]-module K(M) is the free module generated by isotopy 

classes of banded links in M including the empty link, ∅, quotiented by the submodule 

generated by the local relations of Figure 1. 

For any u ∈ C \{0}, wesetK(M,u) = K(M) ⊗ Z[A,A −1 ] C, where A acts on C 

by multiplication by u.


Figure 1. Kauffman relations 

The following proposition is a consequence of the construction of the TQFT. 

PROPOSITION 2.3 ([BH+2, Proposition 1.9]) 

Let M be an oriented connected 3-manifold with p 1 -structure and boundary (without 

boundary or marked points). Then there is a surjective map from K(M,−e iπ/p ) 

to V p (). 

This map is defined by sending the element L ⊗ 1 to Z p (M,L), whereL is 

considered as a banded link with color 1. 

3. Convergence of TQFT 

3.1. Settings 

Let be a closed oriented surface of genus g with p 1 -structure. We denote by Ɣ g the 

mapping class group of .Fixp = 2r. 

If h is an element of Ɣ g , we can construct a cobordism C h from to itself as 

× [0, 1], where we identify the first boundary component with using the identity 

and the second one using h. Ifh ′ is another element of Ɣ g , the cobordisms C h ◦ C h ′ 

and C hh ′ are diffeomorphic. We should obtain a representation of Ɣ g on V p () by 

considering the linear map Z p (C h ). The problem is that we have not chosen any 

p 1 -structure on C h , and we cannot make a canonical choice. 

One way to get rid of this annoying fact is to consider the action of Ɣ g on V p ()⊗ 

V p () ∗ = V p (∐ − ). The action of h on this space is given by Z p (C h ∐ − C h ), 

where we choose any p 1 -structure on C h and put the same one on −C h . The action 

does not depend any further on the p 1 -structure: in fact, if we change the p 1 -structure 

in a cobordism M, the linear map Z p (M) is changed by a multiple of κ, a root of 

unity. When we take the dual, the root becomes its conjugate. Hence the two anomalies 

cancel, and we get a true representation of Ɣ g . 

We thus obtain a sequence of representations (V p () ⊗ V p () ∗ ,Z p ⊗ Zp ∗) of 

Ɣ g , and we want to find their limit in some sense. The problem is that the spaces on


which the mapping class group acts are a priori completely different. We need a way 

to compare them; this is suggested by Proposition 2.3. 

Given a multicurve γ in , one can give it a banded structure by taking a 

neighborhood of it in . We can consider the curve γ as a banded link in × [0, 1] 

by sending it to γ ×{1/2}. We use the same notation for the multicurve on and its 

associated banded link in × [0, 1]. 

In [PS], it is shown that the Kauffman skein module K( × [0, 1]) is a free 

Z[A, A −1 ]-module with a basis of the isotopy classes of multicurves. It provides an 

isomorphism of vector spaces between C() and K( ×[0, 1],u) for any u in C\{0}. 

In particular, using Proposition 2.3, we get a surjective map 

ϕ p : C() ∼ → K( × [0, 1], −e iπ/p ) → V p (∐ − ). 

THEOREM 3.1 

Let be a closed oriented surface of genus g. There is a Hermitian pairing 〈·, ·〉 on 

C() such that for all x and y in C(), the following holds, where d(1) = 1 and 

d(g) = 3g − 3 for g>1: 

2 

) d(g)〈ϕp 

〈x,y〉= lim (x),ϕ 

p→∞( 

p (y)〉 p . 

p 

3.2. The trace function 


Let be a closed oriented surface of genus g, andletγ be a multicurve on . We 

set Tr p (γ ) =〈 × S 1 ,γ〉 p . Here, γ is seen as a banded link with color 1 lying in the 

slice ×{1/2} of × S 1 . 

LEMMA 3.3 

Suppose that a surface is presented as the boundary of a handlebody H which 

retracts on a trivalent banded graph G as in Theorem 2.1. We choose meridian discs 

D e transverse to each edge of G and define C e = ∂D e ; the curves C e are disjoint on 

. We choose a nonnegative integer m e for each edge of G. 

Then we define γ as the multicurve on obtained by taking m e parallel copies 

of C e for each edge of G. We have 

Tr p (γ ) = ∑ σ 

∏ 

e 

[ ( (σe + 1)π 

−2 cos 

r 

)] me. 

Here, σ ranges over r-admissible colorings of G, and e ranges over edges of G.


Figure 2. Action of a curve in TQFT 

Proof 

The proof is an easy consequence of the following fact from skein theory: a trivial 

curve colored with 1 and making a Hopf link with a curve colored with j may be 

removed and replaced by a factor −A 2j+2 −A −2j−2 =−2 cos((j + 1)π/r). We refer, 

for instance, to [BH+1, Lemma 3.2]. 

We use the general trace formula of TQFT (see, e.g., [BH+2, (1.2)]). Let M 

be a cobordism from to , andletƔ be a colored banded graph in M. LetM 

be the closed manifold obtained from M by identifying the two copies of . Then 

〈M ,Ɣ〉 p = Tr Z p (M,Ɣ). 

Consider the basis u σ = Z p (H,G σ ) of V p () involved in Theorem 2.1. For 

each curve C e , the cobordism ( × [0, 1],C e ) acts on u σ by multiplication with 

−2 cos((σ e + 1)π/r), as suggested in Figure 2. 

∏ 

e 

( 

Then the cobordism ( × [0, 1],γ) acts on u σ by multiplication with 

−2 cos((σe + 1)π/r) ) m e. The formula for Trp (γ ) comes now from the trace 

formula of TQFT. 

3.3. Limit of the trace function 

As before, fix a surface , presented as in Theorem 2.1, as the boundary of a handlebody 

H which retracts on a trivalent banded graph G. 

The number of edges of G is 3g − 3 if g>1 or 1 if g = 1. We denote this 

number by d(g) and consider the subset U g of R d(g) consisting of all maps τ from the 

set of edges of G to [0, 1] such that for all triples of incoming edges (e, f, g) of some 

vertex, the following relations are satisfied: 

−|τ f − τ g |≤τ e ≤ τ f + τ g , 

−τ e + τ f + τ g ≤ 2. 

We use the formula of Lemma 3.3 to deduce the asymptotics of the trace function. 

LEMMA 3.4 

With the same hypothesis as in Lemma 3.3, letF γ : U g → R be the map defined by 

F γ (τ) = ∏ e (−2 cos(τ eπ)) m e 

. Then the following formula holds: 

2 

) d(g) 

∫ 

lim Trp (γ ) = 2 

r→∞( g−d(g) F γ (τ) dτ. 

p 

U g


Proof 

The formula for Tr p (γ ) looks like a Riemannian sum; hence the result is not a surprise. 

To obtain the precise result, we have to decompose U g into small pieces parametrized 

by r-admissible colorings σ . 

Given a positive integer r and any coloring σ from the set of edges of G to C r , 

we define the set A r σ 

= ∏ e [σ e/r,σ e + 1/r) ⊂ R d(g) .Asσ runs over r-admissible 

colorings of G, these sets do not cover U g because of the parity condition. We have to 

pack some sets A r σ 

together, which we do in the following way. 

We denote by C 1 (G, Z 2 ) the Z 2 vector space of 1-cochains of G with Z 2 - 

coefficients. The subspace of 1-cycles is denoted by Z 1 (G, Z 2 ). Choose a subspace 

S of C 1 (G, Z 2 ) so that C 1 (G, Z 2 ) = S ⊕ Z 1 (G, Z 2 ). The subspace S has dimension 

d(g) − g. For an admissible coloring σ of G, wedefineBσ r = ⋃ ρ∈S Ar σ +ρ 

. Here, we 

have identified Z/2Z with the set {0, 1}. The sets Bσ r are disjoint and almost cover 

U g . 

Let us prove that they are disjoint. Suppose that we have σ + ρ = σ ′ + ρ ′ with σ 

and σ ′ admissible and ρ,ρ ′ in S; then consider these maps modulo 2. If we apply the 

boundary map, the admissible colorings vanish by definition, and we have ∂ρ = ∂ρ ′ . 

But ∂ induces a bijection from S onto its image; hence we have ρ = ρ ′ , and it follows 

that σ = σ ′ . Hence the sets Bσ r are actually disjoint. Moreover, the measure of Br σ is 

2 d(g)−g /r d(g) . It follows that ∑ ∫ 

σ,r−admissible F γ (σ e + 1/r)(2 g−d(g) /r d(g) ) converges to 

U g 

F γ (τ) dτ, and the result is proved. 

 

3.4. Proof of Theorem 3.1 

Let be a closed oriented surface of genus g. We recall that C() and K(×[0, 1],u) 

are isomorphic as vector spaces for any u in C \{0}. The stacking product induces on 

K( × [0, 1]) a natural algebra structure that induces an algebra structure on C() 

for each u ∈ C \{0}. We consider the algebra structure obtained for u =−1. 

Fix γ and δ, two multicurves on . We aim to compute the limit of the sequence 

(2/p) d(g) 〈ϕ p (γ ),ϕ p (δ)〉 p as p goes to infinity. The right-hand side is the quantum 

invariant of two thickened surfaces with a multicurve inside, glued along their 

boundary. Instead of gluing the two boundaries simultaneously, we glue one and then 

the other. If we glue one boundary component, we obtain the stacking product of γ 

and δ. In the skein module for generic A, we have a decomposition γ · δ = ∑ i c iζ i for 

some multicurves ζ i and some Laurent polynomials c i in Z[A, A −1 ]. When evaluating 

this combination in V p (∐ − ), wehavetospecializeA to −e iπ/p . In formulas, 

we have ϕ p (γ · δ) = ∑ i c i(−e iπ/p )ϕ p (ζ i ). Then, we glue together the remaining 

boundary components and obtain 〈ϕ p (γ ),ϕ p (δ)〉 p = ∑ i c i(−e iπ/p )Tr p (ζ i ). 

The asymptotic formula becomes clear if we define the following linear form on 

C().



Let γ be a multicurve on . Then, there is a pants decomposition associated to γ 

such that all components of γ are parallel copies of the boundary circles. We define 

〈γ 〉=2 g−d(g) ∫ U g 

F γ (τ) dτ, where F γ (τ) = ∏ e (−2 cos(τ eπ)) m e 

. The expression of 

〈γ 〉 as a limit shows that this definition does not depend on the pants decomposition. 

We extend 〈·〉 to a linear form on C(). 

Coming back to our computation, we obtain lim p→∞ (2/p) d(g) 〈ϕ p (γ ),ϕ p (δ)〉 p = 

∑ 

i c i(−1)〈ζ i 〉=〈γδ〉. Finally, we define a Hermitian form on C() by the formula 

〈x,y〉 = 〈xy〉, where the product corresponds to the skein module product 

for A = −1, and the conjugation corresponds to conjugation of coefficients 

in C(). We have proved the following result: for all x,y ∈ C(), wehave 

〈x,y〉=lim p→∞ (2/p) d(g) 〈ϕ p (x),ϕ p (y)〉 p . 

4. Geometric interpretation 

The heart of the following geometric interpretation is the theorem of [B, Theorem 10] 

and [PS, Theorem 4.7] stating that the algebra K( × [0, 1], −1) is isomorphic to 

H(), the ring of regular functions on the SL(2, C)-character variety of . Recall 

that the isomorphism is given by f γ (ρ) =−Tr(ρ(γ )) when γ is a connected curve 

on and ρ : π 1 () → SL(2, C) is a representation of π 1 (). 

This identifies C() with its algebra structure. It remains to identify the linear 

form 〈·〉 of Definition 3.5. 

Recall that the SL(2, C)-character variety contains the SU(2)-character variety, 

which carries a natural symplectic form ω defined in [AB], [G1, page 208]. Following 

[JW, page 154], we define S g to be the moduli space of irreducible representations of 

π 1 () on SU(2) and S g to be the moduli space of all representations. Then it is known 

that S g is a smooth 2d(g)-manifold with symplectic form ω obtained by symplectic 

reduction from the form ω(a,b) = (1/(4π 2 )) ∫ Tr(a ∧ b) for a,b ∈ 1 (,su(2)). 

We denote the volume form on S g by dV = ω d(g) /(d(g)!). 


For all multicurves γ on , we have 

∫ 

〈γ 〉= f γ dV. 

S g 

Proof 

We give a proof of this proposition by adapting the results of [JW]. 

Fix a pants decomposition of associated to γ , and denote the set of curves 

bounding the pants by C e . We define the functions h e on S g with values in [0, 1] by 

the formula Tr ρ(C e ) = 2 cos(πh e (ρ)).


Where the functions h e are not equal to 0 or 1, they Poisson commute, and their 

Hamiltonian flows define a torus action on S g . In fact, we have the following theorem. 

THEOREM 4.2 ([JW, Propositions 3.8, 4.1]) 

Let Ug 

gen be the interior of U g in R d(g) , and let h = (h 1 ,...,h d(g) ):S g → R d(g) be 

the collection of the h e -functions. 

For x in S g such that h(x) = y ∈ Ug 

gen , the torus action identifies h −1 (y) 

with U(1) d(g) /Z 2g−2 

2 , where an element (ε v ) ∈ Z 2g−2 

2 acts on U(1) d(g) by the formula 

e 2iπx e ↦→ (−1)ε v+ε v′ 

e 2iπx e 

,wherev and v ′ are the indices of the pants bounding C e . 

If we choose a Lagrangian submanifold L of S g transverse to the fibers of the 

torus action and which h maps diffeomorphically onto V ⊂ Ug 

gen , then we can define 

canonical coordinates on h −1 (V ) by setting x e = 0 on L and y e = h e . 

The volume form is given on h −1 (V ) by ∏ ∏ 

dy e dxe . 

We come back to the integral of the function associated to γ on the moduli space 

S g . Recall that γ was adapted to the pants decomposition. This means that γ is the 

union of parallel curves C e with multiplicity m e . The function f γ is then defined by 

f γ (ρ) = ∏ e (− Tr ρ(C e)) m e 

= ∏ ( 

e −2 cos(πhe (ρ)) ) m e 

= Fγ (h), where F γ is the 

function of Lemma 3.4. 

As this function depends only on the values of h, we can perform the integration 

on its fiber first. The fiber is isomorphic to U(1) d(g) /Z 2g−2 

2 . Hence U(1) d(g) is a 

Riemannian covering over the fiber and has volume equal to 1. To find the volume 

of the fiber, it is then sufficient to find the degree of this covering. Let G be the 

graph associated to the pants decomposition. The degree of the covering is equal to 

the dimension of the Z 2 -subspace of C 1 (G, Z 2 ) generated by the family of vectors 

u v = e a + e b + e c for each pant v bounding circles a,b, andc. This subspace is 

the image of the coboundary map d : C 0 (G, Z 2 ) → C 1 (G, Z 2 ). Its dimension is 

then complementary to the dimension of H 1 (G, Z 2 ),whichisg. We find that the 

dimension is d(g) − g; hence the covering has degree 2 d(g)−g , and the volume of the 

fiber is 2 g−d(g) . 

We finally obtain ∫ S g 

f γ dV = 2 ∫ g−d(g) U g 

F γ (τ) dτ =〈γ 〉, which completes the 

proof. 

 

Acknowledgment. We thank Gregor Masbaum for his remarks, encouragement, and 

simplification of the proof of Lemma 3.3. 

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[RT] 

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Marché 

Université Pierre et Marie Curie, Analyse Algébrique, Institut de Mathematiques de Jussieu, 

F-75252 Paris CEDEX 05, France; marche@math.jussieu.fr 

Narimannejad 

Université Denis Diderot, Topologie et Géométrie Algébriques, Institut de Mathematiques de 

Jussieu, F-75251 Paris CEDEX 05, France; nariman@math.jussieu.fr; current: Institut für 

Mathematik, Universität Zürich, CH-8057 Zürich, Switzerland; 

majid.narimannejad@math.unizh.ch

A NULLSTELLENSATZ FOR AMOEBAS

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