[PPT] - Random Dieudonn e Modules and the Cohen-Lenstra Heuristics David PowerPoint Presentation

SLIDE 1

Random Dieudonn´ e 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

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Basic Question How often does p divide h(−D)?

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

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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 Dieudonn´ e Modules November 13, 2012 3 / 29

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Guess: Random Integer? P(p | h(−D)) = P(p | D) = 1 p ???

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

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Data (Buell ’76)

P(p | h(−D)) ≈ 1 p + 1 p2 − 1 p5 − 1 p7 + · · · (p odd ) = 1 −

i≥1
1 − 1

pi

= 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 Dieudonn´ e Modules November 13, 2012 5 / 29

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Random finite abelian groups

Idea

P(p | h(−D)) = P(p | #G) = ???

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

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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 Dieudonn´ e Modules November 13, 2012 6 / 29

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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)

G∈G p

1 # Aut G =

i
1 − 1

pi −1 = C −1

p

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

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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)

G∈G p

1 # Aut G =

i
1 − 1

pi −1 = C −1

p

(ii) G → Cp # Aut G is a probability distribution on G p

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

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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)

G∈G p

1 # Aut G =

i
1 − 1

pi −1 = C −1

p

(ii) G → Cp # Aut G is a probability distribution on G p (iii) Avg (#G[p]) = Avg

prp(G)

= 2

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

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Cohen and Lenstra’s conjecture

Let f : G p → Z be a function.

Definition

Avg f =

G∈G p

Cp # Aut G · f (G)

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

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Cohen and Lenstra’s conjecture

Let f : G p → Z be a function.

Definition

Avg f =

G∈G p

Cp # Aut G · f (G) AvgCl f =

0≤D≤X f (Cl(−D)p)
0≤D≤X 1

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

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Cohen and Lenstra’s conjecture

Let f : G p → Z be a function.

Definition

Avg f =

G∈G p

Cp # Aut G · f (G) AvgCl f =

0≤D≤X f (Cl(−D)p)
0≤D≤X 1

Conjecture (Cohen, Lenstra)

(i) AvgCl f = Avg f

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

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Cohen and Lenstra’s conjecture

Let f : G p → Z be a function.

Definition

Avg f =

G∈G p

Cp # Aut G · f (G) AvgCl f =

0≤D≤X f (Cl(−D)p)
0≤D≤X 1

Conjecture (Cohen, Lenstra)

(i) AvgCl f = Avg f (ii) Avg (# Cl(−D)[p]) = 2

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

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Cohen and Lenstra’s conjecture

Let f : G p → Z be a function.

Definition

Avg f =

G∈G p

Cp # Aut G · f (G) AvgCl f =

0≤D≤X f (Cl(−D)p)
0≤D≤X 1

Conjecture (Cohen, Lenstra)

(i) AvgCl f = Avg f (ii) Avg (# Cl(−D)[p]) 2 = 2 + p

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

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Cohen and Lenstra’s conjecture

Let f : G p → Z be a function.

Definition

Avg f =

G∈G p

Cp # Aut G · f (G) AvgCl f =

0≤D≤X f (Cl(−D)p)
0≤D≤X 1

Conjecture (Cohen, Lenstra)

(i) AvgCl f = Avg f (ii) Avg (# Cl(−D)[p]) 2 = 2 + p (iii) P(Cl(−D)p ∼ = G) =

Cp # Aut G .

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

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Progress

Davenport-Heilbronn – Avg Cl(−D)[3] = 2 Bhargava – Avg Cl(K)[2] = 3 (K cubic) Bhargava – counts quartic dihedral extensions Kohnen-Ono – Np ∤h(X) ≫

x

1 2

log x

Heath-Brown – Np|h(X) ≫ x

9 10

log x

Byeon – NClp∼

=(Z/gZ)2(X) ≫ x

1 g

log x

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

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Cohen-Lenstra over Fq(t), ℓ = p Cl(−D) = Pic(Spec OK)

vs

Pic(C)

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

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Cohen-Lenstra over Fq(t), ℓ = p Cl(−D) = Pic(Spec OK)

vs

Pic(C)

deg

− − → Z → 0

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

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Cohen-Lenstra over Fq(t), ℓ = p Cl(−D) = Pic(Spec OK)

vs

0 → Pic0(C) → Pic(C)

deg

− − → Z → 0

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

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Basic Question over Fq(t), ℓ = p

Fix G ∈ G ℓ.

What is

P(Pic0(C)ℓ ∼ = G)? (Limit is taken as deg f → ∞, where C : y2 = f (x).)

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

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Main Tool over Fq(t) – Tate Module Aut Tℓ(JacC) ∼ = Z2g

ℓ

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

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Main Tool over Fq(t) – Tate Module GalFq → Aut Tℓ(JacC) ∼ = Z2g

ℓ

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

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Main Tool over Fq(t) – Tate Module Frob ∈ GalFq → Aut Tℓ(JacC) ∼ = Z2g

ℓ

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

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Main Tool over Fq(t) – Tate Module

Frob ∈ GalFq → Aut Tℓ(JacC) ∼

= Z2g

ℓ

coker (Frob − Id) ∼

= JacC(Fq)ℓ = Pic0(C)

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

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Random Tate-modules

F ∈ GL2g(Zℓ) (w/ Haar measure)

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

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Random Tate-modules

F ∈ GL2g(Zℓ) (w/ Haar measure) Theorem (Friedman, Washington)

P(coker F − I ∼ = L) = Cℓ # Aut L

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

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Random Tate-modules

F ∈ GL2g(Zℓ) (w/ Haar measure) Theorem (Friedman, Washington)

P(coker F − I ∼ = L) = Cℓ # Aut L

Conjecture

P(Pic0(C) ∼ = L) = Cℓ # Aut L

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

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Progress

In the limit (w/ upper and lower densities): Achter – conjectures are true for GSp2g instead of GL2g. Ellenberg-Venkatesh – conjectures are true if ℓ ∤ q − 1. Garton – explicit conjectures for GSp2g, ℓ | q − 1.

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

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Cohen-Lenstra over Fp(t), ℓ = p

Basic question – what is

P(p | # JacC(Fp))?

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

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Cohen-Lenstra over Fp(t), ℓ = p

Tℓ(JacC) ∼ = Zr

ℓ, 0 ≤ r ≤ g

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

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Cohen-Lenstra over Fp(t), ℓ = p

Tℓ(JacC) ∼ = Zr

ℓ, 0 ≤ r ≤ g

Definition

The p-rank of JacC is the integer r.

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

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Cohen-Lenstra over Fp(t), ℓ = p

Tℓ(JacC) ∼ = Zr

ℓ, 0 ≤ r ≤ g

Definition

The p-rank of JacC is the integer r.

Complication

As C varies, r varies. Need to know the distribution of p-ranks, or find a better algebraic gadget than Tℓ(JacC).

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

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Dieudonn´ e Modules

Definition

(i) D = Zq[F, V ]/(FV = VF = p, Fz = zσF, Vz = zσ−1V ).

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

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Dieudonn´ e Modules

Definition

(i) D = Zq[F, V ]/(FV = VF = p, Fz = zσF, Vz = zσ−1V ). (ii) A Dieudonn´ e module is a D-module which is finite and free as a Zq module.

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

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Dieudonn´ e Modules

Definition

(i) D = Zq[F, V ]/(FV = VF = p, Fz = zσF, Vz = zσ−1V ). (ii) A Dieudonn´ e module is a D-module which is finite and free as a Zq module.

JacC

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

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Dieudonn´ e Modules

Definition

(i) D = Zq[F, V ]/(FV = VF = p, Fz = zσF, Vz = zσ−1V ). (ii) A Dieudonn´ e module is a D-module which is finite and free as a Zq module.

M = H1

cris(JacC, Zp)

JacC

David Zureick-Brown (Emory University)

Random Dieudonn´ e Modules November 13, 2012 16 / 29

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Dieudonn´ e Modules

Definition

(i) D = Zq[F, V ]/(FV = VF = p, Fz = zσF, Vz = zσ−1V ). (ii) A Dieudonn´ e module is a D-module which is finite and free as a Zq module.

M = H1

cris(JacC, Zp)

JacC

{JacC[pn]}n
David Zureick-Brown (Emory University)

Random Dieudonn´ e Modules November 13, 2012 16 / 29

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Dieudonn´ e Modules

Definition

(i) D = Zq[F, V ]/(FV = VF = p, Fz = zσF, Vz = zσ−1V ). (ii) A Dieudonn´ e module is a D-module which is finite and free as a Zq module.

M = H1

cris(JacC, Zp)

JacC
H1

dR(JacC, Fp)

{JacC[pn]}n

David Zureick-Brown (Emory University)

Random Dieudonn´ e Modules November 13, 2012 16 / 29

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Dieudonn´ e Modules

Definition

(i) D = Zq[F, V ]/(FV = VF = p, Fz = zσF, Vz = zσ−1V ). (ii) A Dieudonn´ e module is a D-module which is finite and free as a Zq module.

M = H1

cris(JacC, Zp)

JacC
H1

dR(JacC, Fp)

{JacC[pn]}n

V −1: df → “d(f p)”

p

David Zureick-Brown (Emory University)

Random Dieudonn´ e Modules November 13, 2012 16 / 29

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Invariants via Dieudonn´ e Modules

Invariants

(i) p-rank(JacC) = dim F ∞(M ⊗ Fp). (ii) a(JacC) = dim Hom(αp, JacC[p]) = dim (ker V ∩ ker F). (iii) JacC(Fp)p = coker(F − Id)|F ∞(M⊗Fp).

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

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Principally quasi polarized Dieudone´ e modules

Definition

A principally quasi polarized Dieudone´ e module a Dieudone´ e module M together with a non-degenerate symplectic pairing , such that for all x, y ∈ M, Fx, y = σx, Vy.

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

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Main Theorem

Theorem (Cais, Ellenberg, ZB)

(i) Modpqp D has a natural probability measure.

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

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Main Theorem

Theorem (Cais, Ellenberg, ZB)

(i) Modpqp D has a natural probability measure.

(Push forward along Sp2g(Zp)2 → Sp2g(Zp) · F0 · Sp2g(Zp))

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

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Main Theorem

Theorem (Cais, Ellenberg, ZB)

(i) Modpqp D has a natural probability measure.

(Push forward along Sp2g(Zp)2 → Sp2g(Zp) · F0 · Sp2g(Zp))

(ii) P(a(M) = s) = p−(s+1

2 ) ·

∞

i=1
1 + p−i−1 ·

s

i=1
1 − p−i−1 .

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

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Main Theorem

Theorem (Cais, Ellenberg, ZB)

(i) Modpqp D has a natural probability measure.

(Push forward along Sp2g(Zp)2 → Sp2g(Zp) · F0 · Sp2g(Zp))

(ii) P(a(M) = s) = p−(s+1

2 ) ·

∞

i=1
1 + p−i−1 ·

s

i=1
1 − p−i−1 .

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

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

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Main Theorem

Theorem (Cais, Ellenberg, ZB)

(i) Modpqp D has a natural probability measure.

(Push forward along Sp2g(Zp)2 → Sp2g(Zp) · F0 · Sp2g(Zp))

(ii) P(a(M) = s) = p−(s+1

2 ) ·

∞

i=1
1 + p−i−1 ·

s

i=1
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

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

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Main Theorem

Theorem (Cais, Ellenberg, ZB)

(i) Modpqp D has a natural probability measure.

(Push forward along Sp2g(Zp)2 → Sp2g(Zp) · F0 · Sp2g(Zp))

(ii) P(a(M) = s) = p−(s+1

2 ) ·

∞

i=1
1 + p−i−1 ·

s

i=1
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

(iv) 1st moment is 2.

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

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Main Theorem

Theorem (Cais, Ellenberg, ZB)

(i) Modpqp D has a natural probability measure.

(Push forward along Sp2g(Zp)2 → Sp2g(Zp) · F0 · Sp2g(Zp))

(ii) P(a(M) = s) = p−(s+1

2 ) ·

∞

i=1
1 + p−i−1 ·

s

i=1
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

(iv) 1st moment is 2. (v) P

p ∤ # coker(F − Id)|F ∞(M⊗Fp)
= Cp.

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

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Proofs

Part (i)

Modpqp D has a natural probability measure.

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

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Proofs

Part (i)

Modpqp 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 Dieudonn´ e Modules November 13, 2012 20 / 29

SLIDE 52

Proofs

Part (i)

Modpqp D has a natural probability measure.

1 (D, , , F, V ) s.t., FV = VF = p and F(−) , − = σ− , V (−). 2 D = Z2g

q , , =

   I −I   , F0 =    pI I   , V0 = pF −1.

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

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Proofs

Part (i)

Modpqp D has a natural probability measure.

1 (D, , , F, V ) s.t., FV = VF = p and F(−) , − = σ− , V (−). 2 D = Z2g

q , , =

   I −I   , F0 =    pI I   , V0 = pF −1.

Proposition

The double coset space Sp2g(Zp) · F0 · Sp2g(Zp) contains all pqp Dieudone´ e modules.

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

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Proofs

Part (i)

Modpqp D has a natural probability measure.

1 (D, , , F, V ) s.t., FV = VF = p and F(−) , − = σ− , V (−). 2 D = Z2g

q , , =

   I −I   , F0 =    pI I   , V0 = pF −1.

Proposition

The double coset space Sp2g(Zp) · F0 · Sp2g(Zp) contains all pqp Dieudone´ e modules. Proof: Witt’s theorem – Sp2g acts transitively on symplecto-bases.

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

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Proofs

Part (i)

Modpqp D has a natural probability measure.

1 (D, , , F, V ) s.t., FV = VF = p and F(−) , − = σ− , V (−). 2 D = Z2g

q , , =

   I −I   , F0 =    pI I   , V0 = pF −1.

Proposition

The double coset space Sp2g(Zp) · F0 · Sp2g(Zp) contains all pqp Dieudone´ e modules. Proof: Witt’s theorem – Sp2g acts transitively on symplecto-bases. Note: F ∈ Sp2g(Zp), but rather the subset of GSp2g(Zp) of multiplier pg matricies.

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

SLIDE 56

Proofs

Part (ii)

P(a(M) = s) = p−(s+1

2 ) ·

∞

i=1
1 + p−i−1 ·

s

i=1
1 − p−i−1 .

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

SLIDE 57

Proofs

Part (ii)

P(a(M) = s) = p−(s+1

2 ) ·

∞

i=1
1 + p−i−1 ·

s

i=1
1 − p−i−1 .

1 Duality implies that W1 := ker(F ⊗ Fp) and W2 := ker(V ⊗ Fp) are

maximal isotropics.

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

SLIDE 58

Proofs

Part (ii)

P(a(M) = s) = p−(s+1

2 ) ·

∞

i=1
1 + p−i−1 ·

s

i=1
1 − p−i−1 .

1 Duality implies that W1 := ker(F ⊗ Fp) and W2 := ker(V ⊗ Fp) are

maximal isotropics.

2 a(M) = dim (W1 ∩ W2) David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 21 / 29

SLIDE 59

Proofs

Part (ii)

P(a(M) = s) = p−(s+1

2 ) ·

∞

i=1
1 + p−i−1 ·

s

i=1
1 − p−i−1 .

1 Duality implies that W1 := ker(F ⊗ Fp) and W2 := ker(V ⊗ Fp) are

maximal isotropics.

2 a(M) = dim (W1 ∩ W2) 3 Argue that W1 and W2 are randomly distributed. David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 21 / 29

SLIDE 60

Proofs

Part (ii)

P(a(M) = s) = p−(s+1

2 ) ·

∞

i=1
1 + p−i−1 ·

s

i=1
1 − p−i−1 .

1 Duality implies that W1 := ker(F ⊗ Fp) and W2 := ker(V ⊗ Fp) are

maximal isotropics.

2 a(M) = dim (W1 ∩ W2) 3 Argue that W1 and W2 are randomly distributed. 4 This expression is the probability that two random maximal isotropics

intersect with dimension s.

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

SLIDE 61

Proofs

Part (ii)

P(a(M) = s) = p−(s+1

2 ) ·

∞

i=1
1 + p−i−1 ·

s

i=1
1 − p−i−1 .

1 Duality implies that W1 := ker(F ⊗ Fp) and W2 := ker(V ⊗ Fp) are

maximal isotropics.

2 a(M) = dim (W1 ∩ W2) 3 Argue that W1 and W2 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 (Sp2g 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 Dieudonn´ e Modules November 13, 2012 21 / 29

SLIDE 62

Proofs

Part (iii)

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

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

SLIDE 63

Proofs

Part (iii)

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

1 Recall: r(M) = dim F ∞(M) = rank(F ⊗ Fp)g. David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 22 / 29

SLIDE 64

Proofs

Part (iii)

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

1 Recall: r(M) = dim F ∞(M) = rank(F ⊗ Fp)g. 2 (Pr¨

ufer, Crabb, others) The number of nilpotent N ∈ Mn(Fq) is qn(n−1). Able to modify Crabb’s argument:

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

SLIDE 65

Proofs

Part (iii)

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

1 Recall: r(M) = dim F ∞(M) = rank(F ⊗ Fp)g. 2 (Pr¨

ufer, Crabb, others) The number of nilpotent N ∈ Mn(Fq) is qn(n−1). Able to modify Crabb’s argument:

1

Given N nilpotent, get a flag Vi := Ni(V ).

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

SLIDE 66

Proofs

Part (iii)

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

1 Recall: r(M) = dim F ∞(M) = rank(F ⊗ Fp)g. 2 (Pr¨

ufer, Crabb, others) The number of nilpotent N ∈ Mn(Fq) is qn(n−1). Able to modify Crabb’s argument:

1

Given N nilpotent, get a flag Vi := Ni(V ).

2

There is a unique basis {y1, . . . , yg} such that N(yg) = 0 and Vi = Ni(ymi+1), . . . , N(yg−1) (where mi = g − dim Vi−1)

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

SLIDE 67

Proofs

Part (iii)

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

1 Recall: r(M) = dim F ∞(M) = rank(F ⊗ Fp)g. 2 (Pr¨

ufer, Crabb, others) The number of nilpotent N ∈ Mn(Fq) is qn(n−1). Able to modify Crabb’s argument:

1

Given N nilpotent, get a flag Vi := Ni(V ).

2

There is a unique basis {y1, . . . , yg} such that N(yg) = 0 and Vi = Ni(ymi+1), . . . , N(yg−1) (where mi = g − dim Vi−1)

3

The map N → (N(y1), . . . , N(yg−1)) ∈ V n−1 is bijective.

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

SLIDE 68

Proofs

Part (iv)

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

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

SLIDE 69

Proofs

Part (iv)

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

1 First fix the p-corank. David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 23 / 29

SLIDE 70

Proofs

Part (iv)

1st moment is 2: Avg (#G(Fp)[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 Dieudonn´ e Modules November 13, 2012 23 / 29

SLIDE 71

Proofs

Part (iv)

1st moment is 2: Avg (#G(Fp)[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 Dieudonn´ e Modules November 13, 2012 23 / 29

SLIDE 72

Proofs

Part (iv)

1st moment is 2: Avg (#G(Fp)[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 Dieudonn´ e Modules November 13, 2012 23 / 29

SLIDE 73

Proofs

Part (iv)

1st moment is 2: Avg (#G(Fp)[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(Fp) = G et(Fp) = coker(F|Met − Id). David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 23 / 29

SLIDE 74

Proofs

Part (iv)

1st moment is 2: Avg (#G(Fp)[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(Fp) = G et(Fp) = coker(F|Met − Id). 4 F|Met is random in GLg(Zp). David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 23 / 29

SLIDE 75

Proofs

Part (v)

P

p ∤ # coker(F − Id)|F ∞(M⊗Fp)
= Cp.

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

SLIDE 76

Proofs

Part (v)

P

p ∤ # coker(F − Id)|F ∞(M⊗Fp)
= Cp.

Basically the same proof as the last part.

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

SLIDE 77

Data, Moduli Spaces and Wild Speculation

Question

Does P(p ∤ # JacC(Fp)) = Cp?

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

SLIDE 78

Data, Moduli Spaces and Wild Speculation

Question

Does P(p ∤ # JacC(Fp)) = Cp?

Data

C hyperelliptic, p = 2 – YES!

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

SLIDE 79

Data, Moduli Spaces and Wild Speculation

Question

Does P(p ∤ # JacC(Fp)) = Cp?

Data

C hyperelliptic, p = 2 – YES!
C plane curve, p = 2 – YES!

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

SLIDE 80

Data, Moduli Spaces and Wild Speculation

Question

Does P(p ∤ # JacC(Fp)) = Cp?

Data

C hyperelliptic, p = 2 – YES!
C plane curve, p = 2 – YES!
C plane curve, p = 2 –

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

SLIDE 81

Data, Moduli Spaces and Wild Speculation

Question

Does P(p ∤ # JacC(Fp)) = Cp?

Data

C hyperelliptic, p = 2 – YES!
C plane curve, p = 2 – YES!
C plane curve, p = 2 – NO!?!

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

SLIDE 82

C plane curve, p = 2

Theorem (Cais, Ellenberg, ZB)

P(2 ∤ # JacC(F2)) = 0 for plane curves of odd degree.

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

SLIDE 83

C plane curve, p = 2

Theorem (Cais, Ellenberg, ZB)

P(2 ∤ # JacC(F2)) = 0 for plane curves of odd degree. Proof – theta characteristics.

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

SLIDE 84

a-number data

Does

P(a(JacC(Fp)) = 0) =

∞

i=1
1 + p−i−1

=

∞

i=1
1 − p−2i+1

?

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

SLIDE 85

a-number data

Does

P(a(JacC(Fp)) = 0) =

∞

i=1
1 + p−i−1

=

∞

i=1
1 − p−2i+1

?

Data

C hyperelliptic, p = 2 –

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

SLIDE 86

a-number data

Does

P(a(JacC(Fp)) = 0) =

∞

i=1
1 + p−i−1

=

∞

i=1
1 − p−2i+1

?

Data

C hyperelliptic, p = 2 – not quite.

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

SLIDE 87

a-number data

Does

P(a(JacC(Fp)) = 0) =

∞

i=1
1 + p−i−1

=

∞

i=1
1 − p−2i+1

?

Data

C hyperelliptic, p = 2 – not quite.

P(a(JacC(Fp)) = 0) = 1 − 3−1 (p = 3)

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

SLIDE 88

a-number data

Does

P(a(JacC(Fp)) = 0) =

∞

i=1
1 + p−i−1

=

∞

i=1
1 − p−2i+1

?

Data

C hyperelliptic, p = 2 – not quite.

P(a(JacC(Fp)) = 0) = 1 − 3−1 (p = 3) = (1 − 5−1)(1 − 5−3) (p = 5)

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

SLIDE 89

a-number data

Does

P(a(JacC(Fp)) = 0) =

∞

i=1
1 + p−i−1

=

∞

i=1
1 − p−2i+1

?

Data

C hyperelliptic, p = 2 – not quite.

P(a(JacC(Fp)) = 0) = 1 − 3−1 (p = 3) = (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 Dieudonn´ e Modules November 13, 2012 27 / 29

SLIDE 90

Rational points on Moduli Spaces

P(a(JacCf (Fp)) = 0) = limg→∞

#Hord

g (Fp)

#Hg(Fp) .

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

SLIDE 91

Rational points on Moduli Spaces

P(a(JacCf (Fp)) = 0) = limg→∞

#Hord

g (Fp)

#Hg(Fp) .

One can access this through cohomology and the Weil conjectures.

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

SLIDE 92

Rational points on Moduli Spaces

P(a(JacCf (Fp)) = 0) = limg→∞

#Hord

g (Fp)

#Hg(Fp) .

One can access this through cohomology and the Weil conjectures.
Our data suggests that Hord

g

has cohomology that does not arise by pulling back from Hg.

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

SLIDE 93

Rational points on Moduli Spaces

P(a(JacCf (Fp)) = 0) = limg→∞

#Hord

g (Fp)

#Hg(Fp) .

One can access this through cohomology and the Weil conjectures.
Our data suggests that Hord

g

has cohomology that does not arise by pulling back from Hg.

P(a(JacC(Fp)) = 0) = limg→∞

#Mord

g (Fp)

#Mg(Fp) = ???

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

SLIDE 94

Rational points on Moduli Spaces

P(a(JacCf (Fp)) = 0) = limg→∞

#Hord

g (Fp)

#Hg(Fp) .

One can access this through cohomology and the Weil conjectures.
Our data suggests that Hord

g

has cohomology that does not arise by pulling back from Hg.

P(a(JacC(Fp)) = 0) = limg→∞

#Mord

g (Fp)

#Mg(Fp) = ???

P(a(A(Fp)) = 0) = limg→∞

#Aord

g (Fp)

#Ag(Fp) = ???

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

SLIDE 95

Thank you

Thank You!

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