David Zureick-Brown - Rational points and algebraic cycles

Random Dieudonné Modules and the Cohen-Lenstra 

Heuristics 

David Zureick-Brown 

Bryden Cais 

Jordan Ellenberg 

Emory University 

Slides available at http://www.mathcs.emory.edu/~dzb/slides/ 

Arithmetic of abelian varieties in families 

Lausanne, Switzerland 

November 13, 2012

Basic Question 

How often does p divide h(−D) 

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 2 / 29

Basic Question 

What is 

P(p | h(−D)) = lim 

X →∞ 

#{0 ≤ D ≤ X s.t. p | h(−D)} 

 

#{0 ≤ D ≤ X } 


Guess: Random Integer 

P(p | h(−D)) = P(p | D) = 1 p 


Data (Buell ’76) 

P(p | h(−D)) ≈ 1 p + 1 p 2 − 1 p 5 − 1 + · · · (p odd ) 

p7 = 1 − ∏ (1 − 1 ) 

p i i≥1 

= 0.43 . . . ≠ 1/3 (p = 3) 

= 0.23 . . . ≠ 1/5 (p = 5) 

P(Cl(−D) 3 

∼ = Z/9Z) ≈ 0.070 

P(Cl(−D) 3 

∼ = (Z/3Z) 2 ) ≈ 0.0097 


Random finite abelian groups 

Idea 

P(p | h(−D)) = P(p | #G) = 



Idea 

P(p | h(−D)) = P(p | #G) = 

Let G p be the set of isomorphism classes of finite abelian groups of 

p-power order. 



Idea 

P(p | h(−D)) = P(p | #G) = 



Theorem (Cohen, Lenstra) 

(i) 

∑ 1 

# Aut G = ∏ G∈G p i 

( 

1 − 1 p i ) −1 

= C −1 

p 



Idea 

P(p | h(−D)) = P(p | #G) = 




(i) 

∑ 1 


(ii) G ↦→ 

( 

1 − 1 p i ) −1 

= C −1 

p 

C p 

# Aut G is a probability distribution on G p 



Idea 

P(p | h(−D)) = P(p | #G) = 




(i) 

∑ 1 


(ii) G ↦→ 

( 

1 − 1 p i ) −1 

= C −1 

p 

C p 

# Aut G is a probability distribution on G p 

(iii) Avg (#G[p]) = Avg ( p rp(G)) = 2 


Cohen and Lenstra’s conjecture 

Let f : G p → Z be a function. 

Definition 

Avg f = ∑ 

G∈G p 

C p 

# Aut G · f (G) 




Definition 

Avg f = ∑ 

G∈G p 

C p 

# Aut G · f (G) 

Avg Cl f = 

∑ 

0≤D≤X f (Cl(−D) p) 

∑ 

0≤D≤X 1 




Definition 

Avg f = ∑ 

G∈G p 

C p 

# Aut G · f (G) 

Avg Cl f = 

∑ 

0≤D≤X f (Cl(−D) p) 

∑ 

0≤D≤X 1 

Conjecture (Cohen, Lenstra) 

(i) Avg Cl f = Avg f 




Definition 

Avg f = ∑ 

G∈G p 

C p 

# Aut G · f (G) 

Avg Cl f = 

∑ 

0≤D≤X f (Cl(−D) p) 

∑ 

0≤D≤X 1 



(ii) Avg (# Cl(−D)[p]) = 2 




Definition 

Avg f = ∑ 

G∈G p 

C p 

# Aut G · f (G) 

Avg Cl f = 

∑ 

0≤D≤X f (Cl(−D) p) 

∑ 

0≤D≤X 1 



(ii) Avg (# Cl(−D)[p]) 2 = 2 + p 




Definition 

Avg f = ∑ 

G∈G p 

C p 

# Aut G · f (G) 

Avg Cl f = 

∑ 

0≤D≤X f (Cl(−D) p) 

∑ 

0≤D≤X 1 



(ii) Avg (# Cl(−D)[p]) 2 = 2 + p 

(iii) P(Cl(−D) p 

∼ = G) = 

C p 

# Aut G . 


Progress 

Davenport-Heilbronn – Avg Cl(−D)[3] = 2 

Bhargava – Avg Cl(K)[2] = 3 (K cubic) 

Bhargava – counts quartic dihedral extensions 

Kohnen-Ono – N p ∤h (X ) ≫ x 2 

1 

log x 

Heath-Brown – N p|h (X ) ≫ x 10 

9 

log x 

Byeon – N Clp ∼ =(Z/gZ) 

2(X ) ≫ x 1 g 

log x 


Cohen-Lenstra over F q (t), l ≠ p 

Cl(−D) = Pic(Spec O K ) 

vs 

Pic(C) 




vs 

Pic(C) deg 

−→ Z → 0 




vs 

0 → Pic 0 (C) → Pic(C) deg 

−→ Z → 0 


Basic Question over F q (t), l ≠ p 

Fix G ∈ G l . 

What is 

P(Pic 0 (C) l 

∼ = G) 

(Limit is taken as deg f → ∞, where C : y 2 = f (x).) 


Main Tool over F q (t) – Tate Module 

Aut T l (Jac C ) ∼ = Z 2g 

l 



Gal Fq 

→ Aut T l (Jac C ) ∼ = Z 2g 

l 



Frob ∈ Gal Fq 

→ Aut T l (Jac C ) ∼ = Z 2g 

l 



- Frob ∈ Gal Fq → Aut T l (Jac C ) ∼ = Z 2g 

l 

- coker (Frob − Id) ∼ = Jac C (F q ) l = Pic 0 (C) 


Random Tate-modules 

F ∈ GL 2g (Z l ) (w/ Haar measure) 




Theorem (Friedman, Washington) 

P(coker F − I ∼ = L) = 

C l 

# Aut L 




Theorem (Friedman, Washington) 

Conjecture 

P(coker F − I ∼ = L) = 

P(Pic 0 (C) ∼ = L) = 

C l 

# Aut L 

C l 

# Aut L 


Progress 

In the limit (w/ upper and lower densities): 

Achter – conjectures are true for GSp 2g instead of GL 2g . 

Ellenberg-Venkatesh – conjectures are true if l ∤ q − 1. 

Garton – explicit conjectures for GSp 2g , l | q − 1. 


Cohen-Lenstra over F p (t), l = p 

Basic question – what is 

P(p | # Jac C (F p )) 



T l (Jac C ) ∼ = Z r l , 0 ≤ r ≤ g 



Definition 

The p-rank of Jac C is the integer r. 

T l (Jac C ) ∼ = Z r l , 0 ≤ r ≤ g 



Definition 

The p-rank of Jac C is the integer r. 

Complication 

T l (Jac C ) ∼ = Z r l , 0 ≤ r ≤ g 

As C varies, r varies. Need to know the distribution of p-ranks, or find a 

better algebraic gadget than T l (Jac C ). 


Dieudonné Modules 

Definition 

(i) D = Z q [F , V ]/(FV = VF = p, Fz = z σ F , Vz = z σ−1 V ). 



Definition 


(ii) A Dieudonné module is a D-module which is finite and free as a Z q 

module. 



Definition 



module. 

Jac C 



Definition 



module. 

Jac C 

 

 

M = H 1 cris (Jac C, Z p ) 

 

 

 

 

 



Definition 



module. 

Jac C 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

{Jac C [p n ]} n 



Definition 



module. 

Jac C 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

H 1 dR (Jac C, F p ) 

 

 

 




Definition 



module. 

Jac C 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

H 1 dR (Jac C, F p ) 

 

 

 


 

V −1 : df ↦→ “d(f p )” 

p 


Invariants via Dieudonné Modules 

Invariants 

(i) p-rank(Jac C ) = dim F ∞ (M ⊗ F p ). 

(ii) a(Jac C ) = dim Hom(α p , Jac C [p]) = dim (ker V ∩ ker F ). 

(iii) Jac C (F p ) p = coker(F − Id)| F ∞ (M⊗F p). 


Principally quasi polarized Dieudoneé modules 

Definition 

A principally quasi polarized Dieudoneé module a Dieudoneé module M 

together with a non-degenerate symplectic pairing 〈 , 〉 such that for all 

x, y ∈ M, 

〈Fx, y〉 = σ〈x, Vy〉. 


Main Theorem 

Theorem (Cais, Ellenberg, ZB) 

(i) Mod pqp D has a natural probability measure. 


Main Theorem 



(Push forward along Sp 2g (Z p ) 2 → Sp 2g (Z p ) · F 0 · Sp 2g (Z p )) 


Main Theorem 




(ii) P(a(M) = s) = p −(s+1 2 ) · 

∞∏ ( 

1 + p 

−i ) −1 

· 

i=1 

s∏ ( 

1 − p 

−i ) −1 

. 

i=1 


Main Theorem 




(ii) P(a(M) = s) = p −(s+1 2 ) · 

∞∏ ( 

1 + p 

−i ) −1 

· 

i=1 

s∏ ( 

1 − p 

−i ) −1 

. 

(iii) P(r(M) = g − s) = complicated but explicit expression. 

i=1 


Main Theorem 




(ii) P(a(M) = s) = p −(s+1 2 ) · 

∞∏ ( 

1 + p 

−i ) −1 

· 

i=1 

s∏ ( 

1 − p 

−i ) −1 

. 


(iii’) P(r(M) = g − 2) = (p −2 + p −3 ) · 

i=1 

∞∏ ( 

1 + p 

−i ) −1 

i=1 


Main Theorem 




(ii) P(a(M) = s) = p −(s+1 2 ) · 

∞∏ ( 

1 + p 

−i ) −1 

· 

i=1 

s∏ ( 

1 − p 

−i ) −1 

. 


(iii’) P(r(M) = g − 2) = (p −2 + p −3 ) · 

(iv) 1 st moment is 2. 

i=1 

∞∏ ( 

1 + p 

−i ) −1 

i=1 


Main Theorem 




(ii) P(a(M) = s) = p −(s+1 2 ) · 

∞∏ ( 

1 + p 

−i ) −1 

· 

i=1 

s∏ ( 

1 − p 

−i ) −1 

. 


(iii’) P(r(M) = g − 2) = (p −2 + p −3 ) · 

(iv) 1 st moment is 2. 

i=1 

∞∏ ( 

1 + p 

−i ) −1 

i=1 

(v) P ( p ∤ # coker(F − Id)| F ∞ (M⊗F p)) 

= Cp . 


Proofs 

Part (i) 

Mod pqp D has a natural probability measure. 


Proofs 

Part (i) 


1 (D, 〈 , 〉, F , V ) s.t., FV = VF = p and 〈F (−) , −〉 = σ〈− , V (−)〉. 


Proofs 

Part (i) 



⎡ ⎤ ⎡ ⎤ 

2 D = Z 2g ⎢ 

q , 〈 , 〉 = ⎣ 0 I 

−I 0 

⎥ 

⎦, F 0 = 

⎢ 

⎣ pI 0 

0 I 

⎥ 

⎦, V 0 = pF −1 . 


Proofs 

Part (i) 



⎡ ⎤ ⎡ ⎤ 

2 D = Z 2g ⎢ 

q , 〈 , 〉 = ⎣ 0 I 

−I 0 

Proposition 

⎥ 

⎦, F 0 = 

⎢ 

⎣ pI 0 

0 I 

⎥ 

⎦, V 0 = pF −1 . 

The double coset space Sp 2g (Z p ) · F 0 · Sp 2g (Z p ) contains all pqp 

Dieudoneé modules. 


Proofs 

Part (i) 



⎡ ⎤ ⎡ ⎤ 

2 D = Z 2g ⎢ 

q , 〈 , 〉 = ⎣ 0 I 

−I 0 

Proposition 

⎥ 

⎦, F 0 = 

⎢ 

⎣ pI 0 

0 I 

⎥ 

⎦, V 0 = pF −1 . 



Proof: Witt’s theorem – Sp 2g acts transitively on symplecto-bases. 


Proofs 

Part (i) 



⎡ ⎤ ⎡ ⎤ 

2 D = Z 2g ⎢ 

q , 〈 , 〉 = ⎣ 0 I 

−I 0 

Proposition 

⎥ 

⎦, F 0 = 

⎢ 

⎣ pI 0 

0 I 

⎥ 

⎦, V 0 = pF −1 . 



Proof: Witt’s theorem – Sp 2g acts transitively on symplecto-bases. 

Note: F ∉ Sp 2g (Z p ), but rather the subset of GSp 2g (Z p ) of multiplier p g 

matricies. 


Proofs 

Part (ii) 

P(a(M) = s) = p −(s+1 2 ) · 

∞∏ ( 

1 + p 

−i ) −1 

· 

i=1 

s∏ ( 

1 − p 

−i ) −1 

. 

i=1 


Proofs 

Part (ii) 

P(a(M) = s) = p −(s+1 2 ) · 

∞∏ ( 

1 + p 

−i ) −1 

· 

i=1 

s∏ ( 

1 − p 

−i ) −1 

. 

1 Duality implies that W 1 := ker(F ⊗ F p ) and W 2 := ker(V ⊗ F p ) are 

maximal isotropics. 

i=1 


Proofs 

Part (ii) 

P(a(M) = s) = p −(s+1 2 ) · 

∞∏ ( 

1 + p 

−i ) −1 

· 

i=1 

s∏ ( 

1 − p 

−i ) −1 

. 



2 a(M) = dim (W 1 ∩ W 2 ) 

i=1 


Proofs 

Part (ii) 

P(a(M) = s) = p −(s+1 2 ) · 

∞∏ ( 

1 + p 

−i ) −1 

· 

i=1 

s∏ ( 

1 − p 

−i ) −1 

. 



2 a(M) = dim (W 1 ∩ W 2 ) 

i=1 

3 Argue that W 1 and W 2 are randomly distributed. 


Proofs 

Part (ii) 

P(a(M) = s) = p −(s+1 2 ) · 

∞∏ ( 

1 + p 

−i ) −1 

· 

i=1 

s∏ ( 

1 − p 

−i ) −1 

. 



2 a(M) = dim (W 1 ∩ W 2 ) 

i=1 


4 This expression is the probability that two random maximal isotropics 

intersect with dimension s. 


Proofs 

Part (ii) 

P(a(M) = s) = p −(s+1 2 ) · 

∞∏ ( 

1 + p 

−i ) −1 

· 

i=1 

s∏ ( 

1 − p 

−i ) −1 

. 



2 a(M) = dim (W 1 ∩ W 2 ) 

i=1 


4 This expression is the probability that two random maximal isotropics 

intersect with dimension s. 

5 Compute this with Witt’s theorem (Sp 2g acts transitively on pairs of 

maximal isotropics whose intersection has dimension s), and compute 

explicitly the size of the stabilizers. 


Proofs 

Part (iii) 

P(r(M) = g − s) = complicated but explicit expression. 


Proofs 

Part (iii) 


1 Recall: r(M) = dim F ∞ (M) = rank(F ⊗ F p ) g . 


Proofs 

Part (iii) 



2 (Prüfer, Crabb, others) The number of nilpotent N ∈ M n (F q ) is 

q n(n−1) . Able to modify Crabb’s argument: 


Proofs 

Part (iii) 





1 Given N nilpotent, get a flag V i := N i (V ). 


Proofs 

Part (iii) 






2 There is a unique basis {y 1 , . . . , y g } such that N(y g ) = 0 and 

V i = 〈N i (y mi +1), . . . , N(y g−1 )〉 (where m i = g − dim V i−1 ) 


Proofs 

Part (iii) 






2 There is a unique basis {y 1 , . . . , y g } such that N(y g ) = 0 and 

V i = 〈N i (y mi +1), . . . , N(y g−1 )〉 (where m i = g − dim V i−1 ) 

3 The map N ↦→ (N(y 1 ), . . . , N(y g−1 )) ∈ V n−1 is bijective. 


Proofs 

Part (iv) 

1 st moment is 2: Avg (#G(F p )[p]) = 2 


Proofs 

Part (iv) 


1 First fix the p-corank. 


Proofs 

Part (iv) 



1 Associated p-divisible group decomposes as 

G = G m × G et × G ll . 


Proofs 

Part (iv) 




G = G m × G et × G ll . 

2 Fixing the p-corank fixes the dimension of G ll 


Proofs 

Part (iv) 




G = G m × G et × G ll . 


2 (Show that G random ⇒ G et random.) 


Proofs 

Part (iv) 




G = G m × G et × G ll . 



3 G(F p ) = G et (F p ) = coker(F | M et − Id). 


Proofs 

Part (iv) 




G = G m × G et × G ll . 



3 G(F p ) = G et (F p ) = coker(F | M et − Id). 

4 F | M et is random in GL g (Z p ). 


Proofs 

Part (v) 

P ( p ∤ # coker(F − Id)| F ∞ (M⊗F p)) 

= Cp . 


Proofs 

Part (v) 

P ( p ∤ # coker(F − Id)| F ∞ (M⊗F p)) 

= Cp . 

Basically the same proof as the last part. 


Data, Moduli Spaces and Wild Speculation 

Question 

Does P(p ∤ # Jac C (F p )) = C p 



Question 


Data 

- C hyperelliptic, p ≠ 2 – YES! 



Question 


Data 


- C plane curve, p ≠ 2 – YES! 



Question 


Data 



- C plane curve, p = 2 – 



Question 


Data 



- C plane curve, p = 2 – NO!! 


C plane curve, p = 2 


P(2 ∤ # Jac C (F 2 )) = 0 for plane curves of odd degree. 


C plane curve, p = 2 


P(2 ∤ # Jac C (F 2 )) = 0 for plane curves of odd degree. 

Proof – theta characteristics. 


a-number data 

Does 

∞∏ ( 

P(a(Jac C (F p )) = 0) = 1 + p 

−i ) −1 

i=1 

∞∏ ( 

= 1 − p 

−2i+1 ) 

i=1 


a-number data 

Does 

P(a(Jac C (F p )) = 0) = 

Data 

- C hyperelliptic, p ≠ 2 – 

= 

∞∏ ( 

1 + p 

−i ) −1 

i=1 

∞∏ ( 

1 − p 

−2i+1 ) 

i=1 


a-number data 

Does 

P(a(Jac C (F p )) = 0) = 

Data 

- C hyperelliptic, p ≠ 2 – not quite. 

= 

∞∏ ( 

1 + p 

−i ) −1 

i=1 

∞∏ ( 

1 − p 

−2i+1 ) 

i=1 


a-number data 

Does 

P(a(Jac C (F p )) = 0) = 

Data 


= 

∞∏ ( 

1 + p 

−i ) −1 

i=1 

∞∏ ( 

1 − p 

−2i+1 ) 

P(a(Jac C (F p )) = 0) = 1 − 3 −1 (p = 3) 

i=1 


a-number data 

Does 

P(a(Jac C (F p )) = 0) = 

Data 


= 

∞∏ ( 

1 + p 

−i ) −1 

i=1 

∞∏ ( 

1 − p 

−2i+1 ) 

P(a(Jac C (F p )) = 0) = 1 − 3 −1 (p = 3) 

i=1 

= (1 − 5 −1 )(1 − 5 −3 ) (p = 5) 


a-number data 

Does 

P(a(Jac C (F p )) = 0) = 

Data 


= 

∞∏ ( 

1 + p 

−i ) −1 

i=1 

∞∏ ( 

1 − p 

−2i+1 ) 

P(a(Jac C (F p )) = 0) = 1 − 3 −1 (p = 3) 

i=1 

= (1 − 5 −1 )(1 − 5 −3 ) (p = 5) 

= (1 − 7 −1 )(1 − 7 −3 )(1 − 7 −5 ) (p = 7) 


Rational points on Moduli Spaces 

#Hg 

- P(a(Jac Cf (F p )) = 0) = lim ord(Fp) 

g→∞ #H g (F . p) 



#Hg 


g→∞ #H g (F . p) 

- One can access this through cohomology and the Weil conjectures. 



#Hg 


g→∞ #H g (F . p) 


- Our data suggests that Hg 

ord 

pulling back from H g . 

has cohomology that does not arise by 



#Hg 


g→∞ #H g (F . p) 



ord 



- P(a(Jac C (F p )) = 0) = lim g→∞ 

#M ord 

g (F p) 

#M g (F p) 

= 



#Hg 


g→∞ #H g (F . p) 



ord 



- P(a(Jac C (F p )) = 0) = lim g→∞ 

#M ord 

g (F p) 

#M g (F p) 

= 

#A 

- P(a(A(F p )) = 0) = lim ord 

g (Fp) 

g→∞ #A g (F p) 

= 


Thank you 

Thank You!

David Zureick-Brown - Rational points and algebraic cycles

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