Lecture Notes of David Goss

Vanderbilt, May 2009 (three talks)I. General Set-upDavid GossLet X be a geometrically connected smooth projective curve over F q , q = p n 0.Pick a fixed closed point ∞ of X of degree d ∞ . SetA := Γ(X − ∞)= ring of functions holomorphic outside ∞ .For an ideal I of A, set deg (I) := deg FqA/I.Facts:1. A ∗ = F ∗ q,2. A is finitely generated over F q ,3. A is a Dedekind domain with finite class group.We set k to be the quotient field of A and K := k ∞ = completion of k at∞. Let K be a fixed algebraic closure of K with its canonical absolute valueand let C ∞ be the completion of K.Example 1. A = F q [t], k = F q (t), K = F q(( 1t)).N.B.: A is discrete in K and K/A is compact. (Think of Z ⊂ R.)Definition 1. We setS ∞ := C ∗ ∞ × Z p .We write the group operation on S ∞ additively.Next we shall need a notion of positivity which generalizes the notion of“monic polynomial”. Put q ∞ := q d∞ and let F ∞ ≃ F q∞ be the subfield ofconstants of K. A sign function sgn: K ∗ → F ∗ ∞ is a homomorphism which isthe identity on F ∗ ∞.1

Construction: Let π be an element of K with v ∞ (π) = 1, where v ∞ is thecanonical additive valuation at ∞ (i.e., π is a uniformizer). Let x ∈ K ∗ .Havex = ζ x π a 〈x〉where a ∈ Z, ζ x ∈ F ∗ ∞ and 〈x〉 ≡ 1 (π). Setsgn(x) := ζ x .Definition 2. We say x is positive if and only if sgn(x) = 1.The positive elements form a subgroup of K ∗ ; if x is positive and v ∞ (y) >v ∞ (x), then x + y is positive.Example 2. A = F q [t], π = 1 tgives the usual notion of “monic polynomial”.Let u = 1 + w where v ∞ (w) > 0. Let y ∈ Z p . Then u y = (1 + w) y =∞∑ )w j converges in K with the usual properties.j=0( yjMotivating Example: Put A = F q [t]; π = 1 t . Let s = (x, y) ∈ S ∞. We seta s := x −v∞(a) 〈a〉 y= x deg a (a/t deg a ) y .In particular, put s i := (t i , i) = (π −i , i), i ∈ Z. Soa s i= t i deg a (a/t deg a ) i = a i ,in the usual sense of a i .Note that the integral powers lie discretely in S ∞ .For general A we have ideals which are not principal or positively generatedif they are principal; thus have need to “exponentiate” general ideals.For x ∈ K set deg x := −d ∞ v ∞ (x). Let I be the ideal group of A. LetP ⊆ I be the subgroup of principal ideals and P + be subgroup of principaland positively generated ideals. We know I/P + is a finite abelian group.Lemma 1. Let Û1 ⊂ C ∗ ∞ be the group of 1-units (i.e., u = 1 + w wherev ∞ (w) > 0 for w ∈ C ∞ and v ∞ the canonical extension). Then Û1 is aQ p -vector space.Proof. Û 1 is a Z p -module via the binomial theorem. But one can take uniquep j -th roots in characteristic p.2

Have a homomorphism 〈unique positive generator.〉: P + → Û1 given by 〈(a)〉 = 〈a〉 for a theCorollary 1. 〈 〉 extends uniquely to a homomorphism I → Û1.Proof. Û 1 is divisible (and so injective) and I/P + is finite.Definition 3. Let I be an A-ideal and let s = (x, y) ∈ S ∞ . We setI s := x deg I 〈I〉 y .Let π ∗ be a fixed root of u d∞ − π = 0.Definition 4. We set s i := (π∗−i , i).Properties:1. (IJ) s = I s J s2. Let s, z ∈ S ∞ . ThenI s I z = I s+z .3. Let a ∈ A be positive and i ∈ Z. Then(a) s i= a i .Let I be an ideal. Suppose I t = (i) for i ∈ A positive and t > 0 a positiveinteger (in the usual sense). Set z := I (1) := I s 1.Lemma 2. z t = i.Proof. We havez t = (I s 1) t (i.e., the element I s 1raised to t-th power)= I st= (I t ) s 1(i.e., the ideal I t raised to s 1 -power)= (i) s 1(i.e., the principal ideal (i) to the s 1 -power)= i .Definition 5. We set V := k({I s 1}) when I runs over all ideals of A.3

As I/P + is finite, we have [V : k] < ∞.Let O ⊂ V be the A-integers.Proposition 1. Every ideal of A becomes principal in O.Proof. Suppose I t = (i) as above, and let z = I s 1. We have (O · I) t = O · iandz t = i .Comparing orders gives O · z = O · I.4

II. L-seriesMain Idea: Define L-series by Euler products in usual fashion. We willwork with Drinfeld modules and give here a short introduction to them.Let L be a field lying over Spec(A); i.e., there is a map ı: A → L. (Thekernel is called the “characteristic” of L.) Let τ be the endomorphism x ↦→ x qof G a /L. Let L{τ} be the noncommutative algebra of “polynomials” in τ.Definition 6. A Drinfeld module is a homomorphism ϕ: A → L{τ} suchthat1. For a ∈ A, ϕ a (τ) = ı(a)τ 0 + {higher terms}.2. For some a ∈ A, ϕ a (τ) ≠ ı(a)τ 0 .Part 2 is a nontriviality assumption.Let f(τ) =t ∑j=0a j τ j , a t ≠ 0, be in L{τ}. We set deg τ f := t. If ϕ is aDrinfeld module, as above, then for a ∈ A we setv ϕ (a) := − deg τ ϕ a (τ) .The following result then follows easily.Lemma 3.1. v ϕ (ab) = v ϕ (a) + v ϕ (b).2. v ϕ (a + b) ≥ min{v ϕ (a), v ϕ (b)}.Corollary 2. There is a rational number r ϕ such thatv ϕ (a) = r ϕ d ∞ v ∞ (a) = −r ϕ deg a .A simple counting argument shows that r ϕ is a positive integer called therank of the Drinfeld module.Let v be a prime of A not divisible by kernel ı. Let L be an algebraicclosure of L.Definition 7. We set ϕ[v j ] := {α ∈ L | ϕ a (α) = 0 for all a ∈ v j }.(This is the set of v j -division points.) Note that ϕ[v j ] becomes an A-module via ϕ. Counting also gives the next result.5

Lemma 4. ϕ[v j ] ≃ (A/v j ) rϕ as A-modules.Dualizing, we have the Tate module T v (ϕ) which is isomorphic to A rϕv asA v -module.Suppose now that L is a finite field; say L ≃ F q d. Let F := τ d . ClearlyF commutes with ϕ a (τ) for any a ∈ A (coefficients of ϕ a (τ) are fixed by F ).Thus F gives rise to an endomorphism of T v (ϕ).Theorem 1. The characteristic polynomial of F on T v (ϕ) has A-coefficientsindependent of v. Let α ∈ K be a root of P (x). Then|α| = (q d ) 1/rϕ .There are special rank 1 Drinfeld modules called “Hayes-modules”; theyplay the role in the theory classically played by the algebraic group G m . TheHayes-modules are defined over an extension H + of k which is Galois withgroup I/P + in line with class field theory.Let B be a prime of the A-integers of H + . By class field theory, the normof B to A is principal and positively generated. We let nB be the uniquepositive generator of this ideal.Let L/H + be any finite extension of H + . Let I be an A-ideal and letL(I) be the extension adjoining I-division points of a Hayes module (definedabove for I = v j ). We know that L(I)/L is abelian with group isomorphicto a subgroup of A/I ∗ via the action of the Galois group and A on the I-division points. Let B now be a prime of L lying above Spec(A) − V (I); letσ B ∈ Gal(L(I)/L) be the Frobenius morphism. Then we haveσ B ↔ nB + I .Example 3. A = F q [t], L = k = F q (t). Then ϕ is determined byϕ t (τ) = tτ 0 +d∑a i τ i , a d ≠ 0 ;i=0here d is the rank.given byThe simplest Drinfeld module is the “Carlitz module”c t (τ) = tτ 0 + τ .6

Let L/k be finite and let ϕ be a Drinfeld module over L. Let O L ⊂L be the A-integers. As A is finitely generated, the coefficients of ϕ areintegral at almost all primes of O L ; thus we can reduce the Drinfeld moduleat these primes. By throwing out perhaps finitely many other primes, wecan assume that the reduced Drinfeld module has the same rank and so weobtain characteristic polynomials of Frobenius in the usual manner of ellipticcurves.The other (possibly bad) primes are also successfully dealt with in amanner very similar to the theory of elliptic curves.Definition 8. We set L(ϕ, s) := ∏ f B (nB −s ) −1 whereBf B (u) = det (1 − F B u) ,with F B the Frobenius at the good primes. At the bad primes, factors canbe added which are very similar to those arising in the theory of L-series ofelliptic curves.Definition 9. ζ A (s) :=∏ (1 − p −s ) −1 = ∑ I −s .B primeI idealThe estimates on Frobenius eigenvalues implies that these functions convergeon a “half-plane” of S ∞ consisting of those (x, y) such that |x| > c forsome constant c.7

III. Some of What is KnownRecall that we have our finite extension L/k and a Drinfeld module ϕ definedover L. In this situation we have defined L(ϕ, s) (as well as ζ A (s), etc.).Theorem 2. The above functions analytically continue to “entire” twovariablefunctions on S ∞ which are continuous in s = (x, y). Moreover,for each y ∈ Z p we obtain an entire power series L(ϕ; x, y) in x −1 .Proof. Elementary estimates in the case of ζ A (s); cohomology in general case.(See just below.)Such Dirichlet series always have an extremely rich theory of special values;indeed we obtain “special polynomials” as follows. Let i be a nonnegativeinteger and definez(ϕ; x, −i) := L(ϕ; π i ∗x, −i)= Πf B (x − deg nB · nB i ) −1 .For ζ A (s) there is an obvious variant of this definition. One sees easily thatz(ϕ; x, −i) is an entire power series with coefficients in A. Thus, in fact,almost all the coefficients must vanish andz(ϕ; x, −i) ∈ A[x −1 ] ;(for ζ A (s) we obtain a polynomial with coefficients in the integers of the fieldV ). These are the special polynomials.Remarks:1. These polynomials are cohomological (Taguchi, Wan, Böckle, Pink) inthe case of L-series of Drinfeld modules.2. Their degrees (in x −1 ) grow logarithmically with i.3. The coefficients of L(ϕ; x, y), as power series in x −1 , go to 0 uniformlyand exponentially fast.4. The special polynomials interpolate to two-variable entire functions (inthe same sense as above) at all the primes of A.8

Example 4. Let A = F q [t]. Then the special polynomials z(x, −i) associatedto ζ A (s) are∞∑ ( ∑ )z(x, −i) = x −d f i ∈ A[x −1 ] ,d=0f monicdeg f=dwhere the sum in parentheses vanishes for sufficiently large d. It is knownthat1. The zeroes of z(x, −i) are simple and in F q(( 1t)). (In fact, the sameresult is true for all zeroes of ζ A (s); this is the “Riemann hypothesis”for ζ A (s) at ∞ [Wan, Thakur, Diaz-Vargas, Poonen, Sheats].)2. The degree of z(x, −i) in x −1 is an invariant of permuting the q-expansioncoefficients of i (that is, one may arbitrarily take the coefficient of q t inthe q-adic expansion of i and make it the coefficient of q e where t ↦→ eis the effect of some permutation of the nonnegative integers; if j isthe nonnegative integer constructed from i this way, then the degrees ofz(x, −i) and z(x, −j) are equal).3. If i > 0, i ≡ 0 (q − 1), then z(1, −i) = ζ A (−i) = 04. For i negative one obtains a convergent power series and there is a“duality” theorem relating the valuation at ∞ of their coefficients andthose of z(x, −i), i > 0. (Thakur)Remark: Part 2 just above follows from an old nonvanishing criterion of Carlitzfor the coefficients of z(x, −i) which was resurrected by Dinesh Thakur(and essential for Sheats’ work!); the point being that this criterion is visiblyinvariant under permutation of the q-adic digits of i as above.The cohomology of Böckle-Pink is related to the cohomology of coherentsheaves. It has only three (Rf ! , f ∗ , ⊗) of Grothendieck’s six canonicalfunctions and no duality.At the negative integers, we have trivial zeroes arising from cohomologicalfactors at ∞, or for ζ A (s), etc., double convergences. (For p ∈ Spec(A),z(x, −i) modulo pA[x −1 ] “is” a classical L-series modulo p.) This is thegeneralization of Part 3 in the above example to arbitrary base ring A.In general, the trivial zeroes may have order strictly greater than classicallookinglower bounds; such zeroes are called “non-classical”. Calculationsdue to Thakur and Diaz-Vargas, in the case of some ζ-functions, suggest9

these non-classical trivial zeroes have bounded sum of q-adic coefficients andthat these orders are invariant of permuting q-adic coefficients in the mannerdescribed above. Notice that this readily implies that there are only finitelymany orbits of such nonclassical zeroes.For general A there also exists a transcendental λ ∈ C ∞ (analogous to2πi) such that if i > 0, i ≡ 0 (q ∞ − 1) (q ∞ = q d∞ ), one has0 ≠ ζ A (i)/λ i ∈ Twhere T is the composition of V and H + .For A = F q [t] one can form Bernoulli-Carlitz elements (analogous toBernoulli numbers). There is a “von-Staudt” result and one knows that forq > 2, the condition that a prime f of degree d divides the denominatorat i, is invariant under permuting the q d -adic coefficients of i (in the abovemanner). There is a variant of this for q = 2.Keep A = F q [t] for the remainder of this talk. Write ζ A (s) as ζ(x, y) =∞∑x −d f d (−y) whered=0f d (u) :=∑n monicdeg n=d〈n〉 u , u ∈ Z p .The results of Sheats imply that f d (u) = 0 ⇒ u is a nonnegative integer.Moreover, let ρ ∗ (u) now be obtained by simply permuting the q-adic digitsof u, u ∈ Z p , in the above fashion. (This is, in fact, a continuous process onZ p ; we denote the group of such homeomorphisms by S (q) .) Then we havethe basic result thatf d (u) = 0 ⇔ f d (ρ ∗ (u)) = 0 . (1)(In other words, the zero set of f d (u) is stable under S (q) .) Calculations leadus to expect that the set of zeroes of f d (u) also consists of only finitely manyorbits under S (q) .Remarks: Let f(s), g(s) be two entire functions of a complex variable s.Suppose cf(1 − s) = g(s), c ≠ 0. Put F (t) = f(1/2 + it), etc. ThencF (−t) = G(t) and so the Taylor coefficients of F and G at t = 0 live or dietogether.The above remark leads us to look for a nowhere zero function F d,ρ∗ (u) (insteadof a nonzero constant) such thatf d (ρ ∗ (u)) = F d,ρ∗ (u)f d (u) .10

The existence of such a nowhere zero function would then give another proofof (1).Let v be a prime of degree δ in F q [t]. One obtains the v-adic interpolationof ζ A (s) by using the polynomials(1 − v i x −δ )z(x, −i) .The group of permutations associated to v is given by permuting the q δ -adiccoefficients of p-adic numbers (this is just the group S (q δ )). One can see thatpermuting the q-expansion coefficients of an integer by an element of S (q δ )is both continuous p-adically and preserves congruence classes of integersmodulo q δ − 1, as is mandated by the v-adic theory.One can, therefore, ask similar questions v-adically to those given abovein terms of stability of zero sets, nowhere zero functions etc.Note that if we writewhere(1 − v i x −δ )z(x, −i) =f d,v (u) =∑∞∑x −d f d,v (−i)d=0n monicdeg n=d(n,v)=1then f d,v (u) = f d (u) − v u f d−δ (u). Thus the zeroes of f d−δ (u) give rise tozeroes of f d,v (u) (since one knows that f d (u) = 0 ⇒ f d+e (u) = 0 for alle ≥ 0). Therefore, it is reasonable to expect these zeroes give rise to finitelymany orbits under S (q δ ) . Presumably the full set of zeroes would be obtainedby adding on finitely many other orbits (though, unlike at ∞, we do not yeteven know that the zero set of f d,v (u) is stable under S (q )). δFinally, congruences show that the group S (p) naturally occurs as automorphismsof the convolution algebras of characteristic p valued measuresarising in this theory.n u ,11