# David Zureick-Brown - Rational points and algebraic cycles 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 }

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 3 / 29

Guess: Random Integer

P(p | h(−D)) = P(p | D) = 1 p

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 4 / 29

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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 5 / 29

Random finite abelian groups

Idea

P(p | h(−D)) = P(p | #G) =

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 6 / 29

Random finite abelian groups

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.

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 6 / 29

Random finite abelian groups

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.

Theorem (Cohen, Lenstra)

(i)

∑ 1

# Aut G = ∏ G∈G p i

(

1 − 1 p i ) −1

= C −1

p

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 6 / 29

Random finite abelian groups

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.

Theorem (Cohen, Lenstra)

(i)

∑ 1

# Aut G = ∏ G∈G p i

(ii) G ↦→

(

1 − 1 p i ) −1

= C −1

p

C p

# Aut G is a probability distribution on G p

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 6 / 29

Random finite abelian groups

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.

Theorem (Cohen, Lenstra)

(i)

∑ 1

# Aut G = ∏ G∈G p i

(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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 6 / 29

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)

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 7 / 29

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)

Avg Cl f =

0≤D≤X f (Cl(−D) p)

0≤D≤X 1

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 7 / 29

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)

Avg Cl f =

0≤D≤X f (Cl(−D) p)

0≤D≤X 1

Conjecture (Cohen, Lenstra)

(i) Avg Cl f = Avg f

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 7 / 29

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)

Avg Cl f =

0≤D≤X f (Cl(−D) p)

0≤D≤X 1

Conjecture (Cohen, Lenstra)

(i) Avg Cl f = Avg f

(ii) Avg (# Cl(−D)[p]) = 2

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 7 / 29

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)

Avg Cl f =

0≤D≤X f (Cl(−D) p)

0≤D≤X 1

Conjecture (Cohen, Lenstra)

(i) Avg Cl f = Avg f

(ii) Avg (# Cl(−D)[p]) 2 = 2 + p

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 7 / 29

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)

Avg Cl f =

0≤D≤X f (Cl(−D) p)

0≤D≤X 1

Conjecture (Cohen, Lenstra)

(i) Avg Cl f = Avg f

(ii) Avg (# Cl(−D)[p]) 2 = 2 + p

(iii) P(Cl(−D) p

∼ = G) =

C p

# Aut G .

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 7 / 29

Progress

Davenport-Heilbronn – Avg Cl(−D) = 2

Bhargava – Avg Cl(K) = 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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 8 / 29

Cohen-Lenstra over F q (t), l ≠ p

Cl(−D) = Pic(Spec O K )

vs

Pic(C)

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 9 / 29

Cohen-Lenstra over F q (t), l ≠ p

Cl(−D) = Pic(Spec O K )

vs

Pic(C) deg

−→ Z → 0

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 9 / 29

Cohen-Lenstra over F q (t), l ≠ p

Cl(−D) = Pic(Spec O K )

vs

0 → Pic 0 (C) → Pic(C) deg

−→ Z → 0

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 9 / 29

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).)

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 10 / 29

Main Tool over F q (t) – Tate Module

Aut T l (Jac C ) ∼ = Z 2g

l

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 11 / 29

Main Tool over F q (t) – Tate Module

Gal Fq

→ Aut T l (Jac C ) ∼ = Z 2g

l

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 11 / 29

Main Tool over F q (t) – Tate Module

Frob ∈ Gal Fq

→ Aut T l (Jac C ) ∼ = Z 2g

l

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 11 / 29

Main Tool over F q (t) – Tate Module

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

l

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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 11 / 29

Random Tate-modules

F ∈ GL 2g (Z l ) (w/ Haar measure)

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 12 / 29

Random Tate-modules

F ∈ GL 2g (Z l ) (w/ Haar measure)

Theorem (Friedman, Washington)

P(coker F − I ∼ = L) =

C l

# Aut L

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 12 / 29

Random Tate-modules

F ∈ GL 2g (Z l ) (w/ Haar measure)

Theorem (Friedman, Washington)

Conjecture

P(coker F − I ∼ = L) =

P(Pic 0 (C) ∼ = L) =

C l

# Aut L

C l

# Aut L

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 12 / 29

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.

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

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

Basic question – what is

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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 14 / 29

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

T l (Jac C ) ∼ = Z r l , 0 ≤ r ≤ g

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 15 / 29

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

Definition

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

T l (Jac C ) ∼ = Z r l , 0 ≤ r ≤ g

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 15 / 29

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

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 ).

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 15 / 29

Dieudonné Modules

Definition

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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 16 / 29

Dieudonné Modules

Definition

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

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

module.

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 16 / 29

Dieudonné Modules

Definition

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

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

module.

Jac C

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 16 / 29

Dieudonné Modules

Definition

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

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

module.

Jac C

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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 16 / 29

Dieudonné Modules

Definition

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

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

module.

Jac C

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

{Jac C [p n ]} n

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 16 / 29

Dieudonné Modules

Definition

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

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

module.

Jac C

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

H 1 dR (Jac C, F p )

{Jac C [p n ]} n

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 16 / 29

Dieudonné Modules

Definition

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

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

module.

Jac C

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

H 1 dR (Jac C, F p )

{Jac C [p n ]} n

V −1 : df ↦→ “d(f p )”

p

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 16 / 29

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).

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 17 / 29

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〉.

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 18 / 29

Main Theorem

Theorem (Cais, Ellenberg, ZB)

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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 19 / 29

Main Theorem

Theorem (Cais, Ellenberg, ZB)

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

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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 19 / 29

Main Theorem

Theorem (Cais, Ellenberg, ZB)

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

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

(ii) P(a(M) = s) = p −(s+1 2 ) ·

∞∏ (

1 + p

−i ) −1

·

i=1

s∏ (

1 − p

−i ) −1

.

i=1

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 19 / 29

Main Theorem

Theorem (Cais, Ellenberg, ZB)

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

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

(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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 19 / 29

Main Theorem

Theorem (Cais, Ellenberg, ZB)

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

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

(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.

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

i=1

∞∏ (

1 + p

−i ) −1

i=1

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 19 / 29

Main Theorem

Theorem (Cais, Ellenberg, ZB)

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

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

(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.

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

(iv) 1 st moment is 2.

i=1

∞∏ (

1 + p

−i ) −1

i=1

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 19 / 29

Main Theorem

Theorem (Cais, Ellenberg, ZB)

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

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

(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.

(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 .

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 19 / 29

Proofs

Part (i)

Mod pqp D has a natural probability measure.

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 20 / 29

Proofs

Part (i)

Mod pqp D has a natural probability measure.

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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 20 / 29

Proofs

Part (i)

Mod pqp D has a natural probability measure.

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

⎡ ⎤ ⎡ ⎤

2 D = Z 2g ⎢

q , 〈 , 〉 = ⎣ 0 I

−I 0

⎦, F 0 =

⎣ pI 0

0 I

⎦, V 0 = pF −1 .

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 20 / 29

Proofs

Part (i)

Mod pqp D has a natural probability measure.

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

⎡ ⎤ ⎡ ⎤

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.

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 20 / 29

Proofs

Part (i)

Mod pqp D has a natural probability measure.

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

⎡ ⎤ ⎡ ⎤

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.

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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 20 / 29

Proofs

Part (i)

Mod pqp D has a natural probability measure.

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

⎡ ⎤ ⎡ ⎤

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.

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.

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 20 / 29

Proofs

Part (ii)

P(a(M) = s) = p −(s+1 2 ) ·

∞∏ (

1 + p

−i ) −1

·

i=1

s∏ (

1 − p

−i ) −1

.

i=1

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 21 / 29

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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 21 / 29

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.

2 a(M) = dim (W 1 ∩ W 2 )

i=1

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 21 / 29

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.

2 a(M) = dim (W 1 ∩ W 2 )

i=1

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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 21 / 29

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.

2 a(M) = dim (W 1 ∩ W 2 )

i=1

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

4 This expression is the probability that two random maximal isotropics

intersect with dimension s.

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 21 / 29

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.

2 a(M) = dim (W 1 ∩ W 2 )

i=1

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

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.

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 21 / 29

Proofs

Part (iii)

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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 22 / 29

Proofs

Part (iii)

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

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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 22 / 29

Proofs

Part (iii)

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

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

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

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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 22 / 29

Proofs

Part (iii)

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

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

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

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

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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 22 / 29

Proofs

Part (iii)

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

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

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

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

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

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 )

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 22 / 29

Proofs

Part (iii)

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

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

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

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

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

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.

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 22 / 29

Proofs

Part (iv)

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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 23 / 29

Proofs

Part (iv)

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

1 First fix the p-corank.

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 23 / 29

Proofs

Part (iv)

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

1 First fix the p-corank.

1 Associated p-divisible group decomposes as

G = G m × G et × G ll .

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 23 / 29

Proofs

Part (iv)

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

1 First fix the p-corank.

1 Associated p-divisible group decomposes as

G = G m × G et × G ll .

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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 23 / 29

Proofs

Part (iv)

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

1 First fix the p-corank.

1 Associated p-divisible group decomposes as

G = G m × G et × G ll .

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

2 (Show that G random ⇒ G et random.)

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 23 / 29

Proofs

Part (iv)

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

1 First fix the p-corank.

1 Associated p-divisible group decomposes as

G = G m × G et × G ll .

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

2 (Show that G random ⇒ G et random.)

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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 23 / 29

Proofs

Part (iv)

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

1 First fix the p-corank.

1 Associated p-divisible group decomposes as

G = G m × G et × G ll .

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

2 (Show that G random ⇒ G et random.)

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

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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 23 / 29

Proofs

Part (v)

P ( p ∤ # coker(F − Id)| F ∞ (M⊗F p))

= Cp .

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 24 / 29

Proofs

Part (v)

P ( p ∤ # coker(F − Id)| F ∞ (M⊗F p))

= Cp .

Basically the same proof as the last part.

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 24 / 29

Data, Moduli Spaces and Wild Speculation

Question

Does P(p ∤ # Jac C (F p )) = C p

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 25 / 29

Data, Moduli Spaces and Wild Speculation

Question

Does P(p ∤ # Jac C (F p )) = C p

Data

- C hyperelliptic, p ≠ 2 – YES!

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 25 / 29

Data, Moduli Spaces and Wild Speculation

Question

Does P(p ∤ # Jac C (F p )) = C p

Data

- C hyperelliptic, p ≠ 2 – YES!

- C plane curve, p ≠ 2 – YES!

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 25 / 29

Data, Moduli Spaces and Wild Speculation

Question

Does P(p ∤ # Jac C (F p )) = C p

Data

- C hyperelliptic, p ≠ 2 – YES!

- C plane curve, p ≠ 2 – YES!

- C plane curve, p = 2 –

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 25 / 29

Data, Moduli Spaces and Wild Speculation

Question

Does P(p ∤ # Jac C (F p )) = C p

Data

- C hyperelliptic, p ≠ 2 – YES!

- C plane curve, p ≠ 2 – YES!

- C plane curve, p = 2 – NO!!

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 25 / 29

C plane curve, p = 2

Theorem (Cais, Ellenberg, ZB)

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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 26 / 29

C plane curve, p = 2

Theorem (Cais, Ellenberg, ZB)

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

Proof – theta characteristics.

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 26 / 29

a-number data

Does

∞∏ (

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

−i ) −1

i=1

∞∏ (

= 1 − p

−2i+1 )

i=1

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 27 / 29

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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 27 / 29

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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 27 / 29

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 )

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

i=1

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 27 / 29

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 )

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

i=1

= (1 − 5 −1 )(1 − 5 −3 ) (p = 5)

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 27 / 29

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 )

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)

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 27 / 29

Rational points on Moduli Spaces

#Hg

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

g→∞ #H g (F . p)

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 28 / 29

Rational points on Moduli Spaces

#Hg

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

g→∞ #H g (F . p)

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

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 28 / 29

Rational points on Moduli Spaces

#Hg

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

g→∞ #H g (F . p)

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

- Our data suggests that Hg

ord

pulling back from H g .

has cohomology that does not arise by

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 28 / 29

Rational points on Moduli Spaces

#Hg

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

g→∞ #H g (F . p)

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

- Our data suggests that Hg

ord

pulling back from H g .

has cohomology that does not arise by

- P(a(Jac C (F p )) = 0) = lim g→∞

#M ord

g (F p)

#M g (F p)

=

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 28 / 29

Rational points on Moduli Spaces

#Hg

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

g→∞ #H g (F . p)

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

- Our data suggests that Hg

ord

pulling back from H g .

has cohomology that does not arise by

- 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)

=

David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 28 / 29

Thank you

Thank You!

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

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