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# David Zureick-Brown - Rational points and algebraic cycles

David Zureick-Brown - Rational points and algebraic cycles

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R<strong>and</strong>om Dieudonné Modules <strong>and</strong> the Cohen-Lenstra<br />

Heuristics<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong><br />

Bryden Cais<br />

Jordan Ellenberg<br />

Emory University<br />

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

Arithmetic of abelian varieties in families<br />

Lausanne, Switzerl<strong>and</strong><br />

November 13, 2012

Basic Question<br />

How often does p divide h(−D)<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 2 / 29

Basic Question<br />

What is<br />

P(p | h(−D)) = lim<br />

X →∞<br />

#{0 ≤ D ≤ X s.t. p | h(−D)}<br />

<br />

#{0 ≤ D ≤ X }<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 3 / 29

Guess: R<strong>and</strong>om Integer<br />

P(p | h(−D)) = P(p | D) = 1 p <br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 4 / 29

Data (Buell ’76)<br />

P(p | h(−D)) ≈ 1 p + 1 p 2 − 1 p 5 − 1 + · · · (p odd )<br />

p7 = 1 − ∏ (1 − 1 )<br />

p i i≥1<br />

= 0.43 . . . ≠ 1/3 (p = 3)<br />

= 0.23 . . . ≠ 1/5 (p = 5)<br />

P(Cl(−D) 3<br />

∼ = Z/9Z) ≈ 0.070<br />

P(Cl(−D) 3<br />

∼ = (Z/3Z) 2 ) ≈ 0.0097<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 5 / 29

R<strong>and</strong>om finite abelian groups<br />

Idea<br />

P(p | h(−D)) = P(p | #G) = <br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 6 / 29

R<strong>and</strong>om finite abelian groups<br />

Idea<br />

P(p | h(−D)) = P(p | #G) = <br />

Let G p be the set of isomorphism classes of finite abelian groups of<br />

p-power order.<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 6 / 29

R<strong>and</strong>om finite abelian groups<br />

Idea<br />

P(p | h(−D)) = P(p | #G) = <br />

Let G p be the set of isomorphism classes of finite abelian groups of<br />

p-power order.<br />

Theorem (Cohen, Lenstra)<br />

(i)<br />

∑ 1<br />

# Aut G = ∏ G∈G p i<br />

(<br />

1 − 1 p i ) −1<br />

= C −1<br />

p<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 6 / 29

R<strong>and</strong>om finite abelian groups<br />

Idea<br />

P(p | h(−D)) = P(p | #G) = <br />

Let G p be the set of isomorphism classes of finite abelian groups of<br />

p-power order.<br />

Theorem (Cohen, Lenstra)<br />

(i)<br />

∑ 1<br />

# Aut G = ∏ G∈G p i<br />

(ii) G ↦→<br />

(<br />

1 − 1 p i ) −1<br />

= C −1<br />

p<br />

C p<br />

# Aut G is a probability distribution on G p<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 6 / 29

R<strong>and</strong>om finite abelian groups<br />

Idea<br />

P(p | h(−D)) = P(p | #G) = <br />

Let G p be the set of isomorphism classes of finite abelian groups of<br />

p-power order.<br />

Theorem (Cohen, Lenstra)<br />

(i)<br />

∑ 1<br />

# Aut G = ∏ G∈G p i<br />

(ii) G ↦→<br />

(<br />

1 − 1 p i ) −1<br />

= C −1<br />

p<br />

C p<br />

# Aut G is a probability distribution on G p<br />

(iii) Avg (#G[p]) = Avg ( p rp(G)) = 2<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 6 / 29

Cohen <strong>and</strong> Lenstra’s conjecture<br />

Let f : G p → Z be a function.<br />

Definition<br />

Avg f = ∑<br />

G∈G p<br />

C p<br />

# Aut G · f (G)<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 7 / 29

Cohen <strong>and</strong> Lenstra’s conjecture<br />

Let f : G p → Z be a function.<br />

Definition<br />

Avg f = ∑<br />

G∈G p<br />

C p<br />

# Aut G · f (G)<br />

Avg Cl f =<br />

∑<br />

0≤D≤X f (Cl(−D) p)<br />

∑<br />

0≤D≤X 1<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 7 / 29

Cohen <strong>and</strong> Lenstra’s conjecture<br />

Let f : G p → Z be a function.<br />

Definition<br />

Avg f = ∑<br />

G∈G p<br />

C p<br />

# Aut G · f (G)<br />

Avg Cl f =<br />

∑<br />

0≤D≤X f (Cl(−D) p)<br />

∑<br />

0≤D≤X 1<br />

Conjecture (Cohen, Lenstra)<br />

(i) Avg Cl f = Avg f<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 7 / 29

Cohen <strong>and</strong> Lenstra’s conjecture<br />

Let f : G p → Z be a function.<br />

Definition<br />

Avg f = ∑<br />

G∈G p<br />

C p<br />

# Aut G · f (G)<br />

Avg Cl f =<br />

∑<br />

0≤D≤X f (Cl(−D) p)<br />

∑<br />

0≤D≤X 1<br />

Conjecture (Cohen, Lenstra)<br />

(i) Avg Cl f = Avg f<br />

(ii) Avg (# Cl(−D)[p]) = 2<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 7 / 29

Cohen <strong>and</strong> Lenstra’s conjecture<br />

Let f : G p → Z be a function.<br />

Definition<br />

Avg f = ∑<br />

G∈G p<br />

C p<br />

# Aut G · f (G)<br />

Avg Cl f =<br />

∑<br />

0≤D≤X f (Cl(−D) p)<br />

∑<br />

0≤D≤X 1<br />

Conjecture (Cohen, Lenstra)<br />

(i) Avg Cl f = Avg f<br />

(ii) Avg (# Cl(−D)[p]) 2 = 2 + p<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 7 / 29

Cohen <strong>and</strong> Lenstra’s conjecture<br />

Let f : G p → Z be a function.<br />

Definition<br />

Avg f = ∑<br />

G∈G p<br />

C p<br />

# Aut G · f (G)<br />

Avg Cl f =<br />

∑<br />

0≤D≤X f (Cl(−D) p)<br />

∑<br />

0≤D≤X 1<br />

Conjecture (Cohen, Lenstra)<br />

(i) Avg Cl f = Avg f<br />

(ii) Avg (# Cl(−D)[p]) 2 = 2 + p<br />

(iii) P(Cl(−D) p<br />

∼ = G) =<br />

C p<br />

# Aut G .<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 7 / 29

Progress<br />

Davenport-Heilbronn – Avg Cl(−D)[3] = 2<br />

Bhargava – Avg Cl(K)[2] = 3 (K cubic)<br />

Bhargava – counts quartic dihedral extensions<br />

Kohnen-Ono – N p ∤h (X ) ≫ x 2<br />

1<br />

log x<br />

Heath-<strong>Brown</strong> – N p|h (X ) ≫ x 10<br />

9<br />

log x<br />

Byeon – N Clp ∼ =(Z/gZ)<br />

2(X ) ≫ x 1 g<br />

log x<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 8 / 29

Cohen-Lenstra over F q (t), l ≠ p<br />

Cl(−D) = Pic(Spec O K )<br />

vs<br />

Pic(C)<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 9 / 29

Cohen-Lenstra over F q (t), l ≠ p<br />

Cl(−D) = Pic(Spec O K )<br />

vs<br />

Pic(C) deg<br />

−→ Z → 0<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 9 / 29

Cohen-Lenstra over F q (t), l ≠ p<br />

Cl(−D) = Pic(Spec O K )<br />

vs<br />

0 → Pic 0 (C) → Pic(C) deg<br />

−→ Z → 0<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 9 / 29

Basic Question over F q (t), l ≠ p<br />

Fix G ∈ G l .<br />

What is<br />

P(Pic 0 (C) l<br />

∼ = G)<br />

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

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 10 / 29

Main Tool over F q (t) – Tate Module<br />

Aut T l (Jac C ) ∼ = Z 2g<br />

l<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 11 / 29

Main Tool over F q (t) – Tate Module<br />

Gal Fq<br />

→ Aut T l (Jac C ) ∼ = Z 2g<br />

l<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 11 / 29

Main Tool over F q (t) – Tate Module<br />

Frob ∈ Gal Fq<br />

→ Aut T l (Jac C ) ∼ = Z 2g<br />

l<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 11 / 29

Main Tool over F q (t) – Tate Module<br />

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

l<br />

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

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 11 / 29

R<strong>and</strong>om Tate-modules<br />

F ∈ GL 2g (Z l ) (w/ Haar measure)<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 12 / 29

R<strong>and</strong>om Tate-modules<br />

F ∈ GL 2g (Z l ) (w/ Haar measure)<br />

Theorem (Friedman, Washington)<br />

P(coker F − I ∼ = L) =<br />

C l<br />

# Aut L<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 12 / 29

R<strong>and</strong>om Tate-modules<br />

F ∈ GL 2g (Z l ) (w/ Haar measure)<br />

Theorem (Friedman, Washington)<br />

Conjecture<br />

P(coker F − I ∼ = L) =<br />

P(Pic 0 (C) ∼ = L) =<br />

C l<br />

# Aut L<br />

C l<br />

# Aut L<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 12 / 29

Progress<br />

In the limit (w/ upper <strong>and</strong> lower densities):<br />

Achter – conjectures are true for GSp 2g instead of GL 2g .<br />

Ellenberg-Venkatesh – conjectures are true if l ∤ q − 1.<br />

Garton – explicit conjectures for GSp 2g , l | q − 1.<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 13 / 29

Cohen-Lenstra over F p (t), l = p<br />

Basic question – what is<br />

P(p | # Jac C (F p ))<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 14 / 29

Cohen-Lenstra over F p (t), l = p<br />

T l (Jac C ) ∼ = Z r l , 0 ≤ r ≤ g<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 15 / 29

Cohen-Lenstra over F p (t), l = p<br />

Definition<br />

The p-rank of Jac C is the integer r.<br />

T l (Jac C ) ∼ = Z r l , 0 ≤ r ≤ g<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 15 / 29

Cohen-Lenstra over F p (t), l = p<br />

Definition<br />

The p-rank of Jac C is the integer r.<br />

Complication<br />

T l (Jac C ) ∼ = Z r l , 0 ≤ r ≤ g<br />

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

better <strong>algebraic</strong> gadget than T l (Jac C ).<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 15 / 29

Dieudonné Modules<br />

Definition<br />

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

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 16 / 29

Dieudonné Modules<br />

Definition<br />

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

(ii) A Dieudonné module is a D-module which is finite <strong>and</strong> free as a Z q<br />

module.<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 16 / 29

Dieudonné Modules<br />

Definition<br />

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

(ii) A Dieudonné module is a D-module which is finite <strong>and</strong> free as a Z q<br />

module.<br />

Jac C<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 16 / 29

Dieudonné Modules<br />

Definition<br />

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

(ii) A Dieudonné module is a D-module which is finite <strong>and</strong> free as a Z q<br />

module.<br />

Jac C<br />

<br />

<br />

M = H 1 cris (Jac C, Z p )<br />

<br />

<br />

<br />

<br />

<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 16 / 29

Dieudonné Modules<br />

Definition<br />

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

(ii) A Dieudonné module is a D-module which is finite <strong>and</strong> free as a Z q<br />

module.<br />

Jac C<br />

M = H 1 cris (Jac C, Z p )<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

{Jac C [p n ]} n<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 16 / 29

Dieudonné Modules<br />

Definition<br />

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

(ii) A Dieudonné module is a D-module which is finite <strong>and</strong> free as a Z q<br />

module.<br />

Jac C<br />

M = H 1 cris (Jac C, Z p )<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

H 1 dR (Jac C, F p )<br />

<br />

<br />

<br />

{Jac C [p n ]} n<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 16 / 29

Dieudonné Modules<br />

Definition<br />

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

(ii) A Dieudonné module is a D-module which is finite <strong>and</strong> free as a Z q<br />

module.<br />

Jac C<br />

M = H 1 cris (Jac C, Z p )<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

H 1 dR (Jac C, F p )<br />

<br />

<br />

<br />

{Jac C [p n ]} n<br />

<br />

V −1 : df ↦→ “d(f p )”<br />

p<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 16 / 29

Invariants via Dieudonné Modules<br />

Invariants<br />

(i) p-rank(Jac C ) = dim F ∞ (M ⊗ F p ).<br />

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

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

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 17 / 29

Principally quasi polarized Dieudoneé modules<br />

Definition<br />

A principally quasi polarized Dieudoneé module a Dieudoneé module M<br />

together with a non-degenerate symplectic pairing 〈 , 〉 such that for all<br />

x, y ∈ M,<br />

〈Fx, y〉 = σ〈x, Vy〉.<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 18 / 29

Main Theorem<br />

Theorem (Cais, Ellenberg, ZB)<br />

(i) Mod pqp D has a natural probability measure.<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 19 / 29

Main Theorem<br />

Theorem (Cais, Ellenberg, ZB)<br />

(i) Mod pqp D has a natural probability measure.<br />

(Push forward along Sp 2g (Z p ) 2 → Sp 2g (Z p ) · F 0 · Sp 2g (Z p ))<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 19 / 29

Main Theorem<br />

Theorem (Cais, Ellenberg, ZB)<br />

(i) Mod pqp D has a natural probability measure.<br />

(Push forward along Sp 2g (Z p ) 2 → Sp 2g (Z p ) · F 0 · Sp 2g (Z p ))<br />

(ii) P(a(M) = s) = p −(s+1 2 ) ·<br />

∞∏ (<br />

1 + p<br />

−i ) −1<br />

·<br />

i=1<br />

s∏ (<br />

1 − p<br />

−i ) −1<br />

.<br />

i=1<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 19 / 29

Main Theorem<br />

Theorem (Cais, Ellenberg, ZB)<br />

(i) Mod pqp D has a natural probability measure.<br />

(Push forward along Sp 2g (Z p ) 2 → Sp 2g (Z p ) · F 0 · Sp 2g (Z p ))<br />

(ii) P(a(M) = s) = p −(s+1 2 ) ·<br />

∞∏ (<br />

1 + p<br />

−i ) −1<br />

·<br />

i=1<br />

s∏ (<br />

1 − p<br />

−i ) −1<br />

.<br />

(iii) P(r(M) = g − s) = complicated but explicit expression.<br />

i=1<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 19 / 29

Main Theorem<br />

Theorem (Cais, Ellenberg, ZB)<br />

(i) Mod pqp D has a natural probability measure.<br />

(Push forward along Sp 2g (Z p ) 2 → Sp 2g (Z p ) · F 0 · Sp 2g (Z p ))<br />

(ii) P(a(M) = s) = p −(s+1 2 ) ·<br />

∞∏ (<br />

1 + p<br />

−i ) −1<br />

·<br />

i=1<br />

s∏ (<br />

1 − p<br />

−i ) −1<br />

.<br />

(iii) P(r(M) = g − s) = complicated but explicit expression.<br />

(iii’) P(r(M) = g − 2) = (p −2 + p −3 ) ·<br />

i=1<br />

∞∏ (<br />

1 + p<br />

−i ) −1<br />

i=1<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 19 / 29

Main Theorem<br />

Theorem (Cais, Ellenberg, ZB)<br />

(i) Mod pqp D has a natural probability measure.<br />

(Push forward along Sp 2g (Z p ) 2 → Sp 2g (Z p ) · F 0 · Sp 2g (Z p ))<br />

(ii) P(a(M) = s) = p −(s+1 2 ) ·<br />

∞∏ (<br />

1 + p<br />

−i ) −1<br />

·<br />

i=1<br />

s∏ (<br />

1 − p<br />

−i ) −1<br />

.<br />

(iii) P(r(M) = g − s) = complicated but explicit expression.<br />

(iii’) P(r(M) = g − 2) = (p −2 + p −3 ) ·<br />

(iv) 1 st moment is 2.<br />

i=1<br />

∞∏ (<br />

1 + p<br />

−i ) −1<br />

i=1<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 19 / 29

Main Theorem<br />

Theorem (Cais, Ellenberg, ZB)<br />

(i) Mod pqp D has a natural probability measure.<br />

(Push forward along Sp 2g (Z p ) 2 → Sp 2g (Z p ) · F 0 · Sp 2g (Z p ))<br />

(ii) P(a(M) = s) = p −(s+1 2 ) ·<br />

∞∏ (<br />

1 + p<br />

−i ) −1<br />

·<br />

i=1<br />

s∏ (<br />

1 − p<br />

−i ) −1<br />

.<br />

(iii) P(r(M) = g − s) = complicated but explicit expression.<br />

(iii’) P(r(M) = g − 2) = (p −2 + p −3 ) ·<br />

(iv) 1 st moment is 2.<br />

i=1<br />

∞∏ (<br />

1 + p<br />

−i ) −1<br />

i=1<br />

(v) P ( p ∤ # coker(F − Id)| F ∞ (M⊗F p))<br />

= Cp .<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 19 / 29

Proofs<br />

Part (i)<br />

Mod pqp D has a natural probability measure.<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 20 / 29

Proofs<br />

Part (i)<br />

Mod pqp D has a natural probability measure.<br />

1 (D, 〈 , 〉, F , V ) s.t., FV = VF = p <strong>and</strong> 〈F (−) , −〉 = σ〈− , V (−)〉.<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 20 / 29

Proofs<br />

Part (i)<br />

Mod pqp D has a natural probability measure.<br />

1 (D, 〈 , 〉, F , V ) s.t., FV = VF = p <strong>and</strong> 〈F (−) , −〉 = σ〈− , V (−)〉.<br />

⎡ ⎤ ⎡ ⎤<br />

2 D = Z 2g ⎢<br />

q , 〈 , 〉 = ⎣ 0 I<br />

−I 0<br />

⎥<br />

⎦, F 0 =<br />

⎢<br />

⎣ pI 0<br />

0 I<br />

⎥<br />

⎦, V 0 = pF −1 .<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 20 / 29

Proofs<br />

Part (i)<br />

Mod pqp D has a natural probability measure.<br />

1 (D, 〈 , 〉, F , V ) s.t., FV = VF = p <strong>and</strong> 〈F (−) , −〉 = σ〈− , V (−)〉.<br />

⎡ ⎤ ⎡ ⎤<br />

2 D = Z 2g ⎢<br />

q , 〈 , 〉 = ⎣ 0 I<br />

−I 0<br />

Proposition<br />

⎥<br />

⎦, F 0 =<br />

⎢<br />

⎣ pI 0<br />

0 I<br />

⎥<br />

⎦, V 0 = pF −1 .<br />

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

Dieudoneé modules.<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 20 / 29

Proofs<br />

Part (i)<br />

Mod pqp D has a natural probability measure.<br />

1 (D, 〈 , 〉, F , V ) s.t., FV = VF = p <strong>and</strong> 〈F (−) , −〉 = σ〈− , V (−)〉.<br />

⎡ ⎤ ⎡ ⎤<br />

2 D = Z 2g ⎢<br />

q , 〈 , 〉 = ⎣ 0 I<br />

−I 0<br />

Proposition<br />

⎥<br />

⎦, F 0 =<br />

⎢<br />

⎣ pI 0<br />

0 I<br />

⎥<br />

⎦, V 0 = pF −1 .<br />

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

Dieudoneé modules.<br />

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

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 20 / 29

Proofs<br />

Part (i)<br />

Mod pqp D has a natural probability measure.<br />

1 (D, 〈 , 〉, F , V ) s.t., FV = VF = p <strong>and</strong> 〈F (−) , −〉 = σ〈− , V (−)〉.<br />

⎡ ⎤ ⎡ ⎤<br />

2 D = Z 2g ⎢<br />

q , 〈 , 〉 = ⎣ 0 I<br />

−I 0<br />

Proposition<br />

⎥<br />

⎦, F 0 =<br />

⎢<br />

⎣ pI 0<br />

0 I<br />

⎥<br />

⎦, V 0 = pF −1 .<br />

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

Dieudoneé modules.<br />

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

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

matricies.<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 20 / 29

Proofs<br />

Part (ii)<br />

P(a(M) = s) = p −(s+1 2 ) ·<br />

∞∏ (<br />

1 + p<br />

−i ) −1<br />

·<br />

i=1<br />

s∏ (<br />

1 − p<br />

−i ) −1<br />

.<br />

i=1<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 21 / 29

Proofs<br />

Part (ii)<br />

P(a(M) = s) = p −(s+1 2 ) ·<br />

∞∏ (<br />

1 + p<br />

−i ) −1<br />

·<br />

i=1<br />

s∏ (<br />

1 − p<br />

−i ) −1<br />

.<br />

1 Duality implies that W 1 := ker(F ⊗ F p ) <strong>and</strong> W 2 := ker(V ⊗ F p ) are<br />

maximal isotropics.<br />

i=1<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 21 / 29

Proofs<br />

Part (ii)<br />

P(a(M) = s) = p −(s+1 2 ) ·<br />

∞∏ (<br />

1 + p<br />

−i ) −1<br />

·<br />

i=1<br />

s∏ (<br />

1 − p<br />

−i ) −1<br />

.<br />

1 Duality implies that W 1 := ker(F ⊗ F p ) <strong>and</strong> W 2 := ker(V ⊗ F p ) are<br />

maximal isotropics.<br />

2 a(M) = dim (W 1 ∩ W 2 )<br />

i=1<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 21 / 29

Proofs<br />

Part (ii)<br />

P(a(M) = s) = p −(s+1 2 ) ·<br />

∞∏ (<br />

1 + p<br />

−i ) −1<br />

·<br />

i=1<br />

s∏ (<br />

1 − p<br />

−i ) −1<br />

.<br />

1 Duality implies that W 1 := ker(F ⊗ F p ) <strong>and</strong> W 2 := ker(V ⊗ F p ) are<br />

maximal isotropics.<br />

2 a(M) = dim (W 1 ∩ W 2 )<br />

i=1<br />

3 Argue that W 1 <strong>and</strong> W 2 are r<strong>and</strong>omly distributed.<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 21 / 29

Proofs<br />

Part (ii)<br />

P(a(M) = s) = p −(s+1 2 ) ·<br />

∞∏ (<br />

1 + p<br />

−i ) −1<br />

·<br />

i=1<br />

s∏ (<br />

1 − p<br />

−i ) −1<br />

.<br />

1 Duality implies that W 1 := ker(F ⊗ F p ) <strong>and</strong> W 2 := ker(V ⊗ F p ) are<br />

maximal isotropics.<br />

2 a(M) = dim (W 1 ∩ W 2 )<br />

i=1<br />

3 Argue that W 1 <strong>and</strong> W 2 are r<strong>and</strong>omly distributed.<br />

4 This expression is the probability that two r<strong>and</strong>om maximal isotropics<br />

intersect with dimension s.<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 21 / 29

Proofs<br />

Part (ii)<br />

P(a(M) = s) = p −(s+1 2 ) ·<br />

∞∏ (<br />

1 + p<br />

−i ) −1<br />

·<br />

i=1<br />

s∏ (<br />

1 − p<br />

−i ) −1<br />

.<br />

1 Duality implies that W 1 := ker(F ⊗ F p ) <strong>and</strong> W 2 := ker(V ⊗ F p ) are<br />

maximal isotropics.<br />

2 a(M) = dim (W 1 ∩ W 2 )<br />

i=1<br />

3 Argue that W 1 <strong>and</strong> W 2 are r<strong>and</strong>omly distributed.<br />

4 This expression is the probability that two r<strong>and</strong>om maximal isotropics<br />

intersect with dimension s.<br />

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

maximal isotropics whose intersection has dimension s), <strong>and</strong> compute<br />

explicitly the size of the stabilizers.<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 21 / 29

Proofs<br />

Part (iii)<br />

P(r(M) = g − s) = complicated but explicit expression.<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 22 / 29

Proofs<br />

Part (iii)<br />

P(r(M) = g − s) = complicated but explicit expression.<br />

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

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 22 / 29

Proofs<br />

Part (iii)<br />

P(r(M) = g − s) = complicated but explicit expression.<br />

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

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

q n(n−1) . Able to modify Crabb’s argument:<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 22 / 29

Proofs<br />

Part (iii)<br />

P(r(M) = g − s) = complicated but explicit expression.<br />

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

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

q n(n−1) . Able to modify Crabb’s argument:<br />

1 Given N nilpotent, get a flag V i := N i (V ).<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 22 / 29

Proofs<br />

Part (iii)<br />

P(r(M) = g − s) = complicated but explicit expression.<br />

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

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

q n(n−1) . Able to modify Crabb’s argument:<br />

1 Given N nilpotent, get a flag V i := N i (V ).<br />

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

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

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 22 / 29

Proofs<br />

Part (iii)<br />

P(r(M) = g − s) = complicated but explicit expression.<br />

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

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

q n(n−1) . Able to modify Crabb’s argument:<br />

1 Given N nilpotent, get a flag V i := N i (V ).<br />

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

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

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

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 22 / 29

Proofs<br />

Part (iv)<br />

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

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 23 / 29

Proofs<br />

Part (iv)<br />

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

1 First fix the p-corank.<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 23 / 29

Proofs<br />

Part (iv)<br />

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

1 First fix the p-corank.<br />

1 Associated p-divisible group decomposes as<br />

G = G m × G et × G ll .<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 23 / 29

Proofs<br />

Part (iv)<br />

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

1 First fix the p-corank.<br />

1 Associated p-divisible group decomposes as<br />

G = G m × G et × G ll .<br />

2 Fixing the p-corank fixes the dimension of G ll<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 23 / 29

Proofs<br />

Part (iv)<br />

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

1 First fix the p-corank.<br />

1 Associated p-divisible group decomposes as<br />

G = G m × G et × G ll .<br />

2 Fixing the p-corank fixes the dimension of G ll<br />

2 (Show that G r<strong>and</strong>om ⇒ G et r<strong>and</strong>om.)<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 23 / 29

Proofs<br />

Part (iv)<br />

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

1 First fix the p-corank.<br />

1 Associated p-divisible group decomposes as<br />

G = G m × G et × G ll .<br />

2 Fixing the p-corank fixes the dimension of G ll<br />

2 (Show that G r<strong>and</strong>om ⇒ G et r<strong>and</strong>om.)<br />

3 G(F p ) = G et (F p ) = coker(F | M et − Id).<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 23 / 29

Proofs<br />

Part (iv)<br />

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

1 First fix the p-corank.<br />

1 Associated p-divisible group decomposes as<br />

G = G m × G et × G ll .<br />

2 Fixing the p-corank fixes the dimension of G ll<br />

2 (Show that G r<strong>and</strong>om ⇒ G et r<strong>and</strong>om.)<br />

3 G(F p ) = G et (F p ) = coker(F | M et − Id).<br />

4 F | M et is r<strong>and</strong>om in GL g (Z p ).<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 23 / 29

Proofs<br />

Part (v)<br />

P ( p ∤ # coker(F − Id)| F ∞ (M⊗F p))<br />

= Cp .<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 24 / 29

Proofs<br />

Part (v)<br />

P ( p ∤ # coker(F − Id)| F ∞ (M⊗F p))<br />

= Cp .<br />

Basically the same proof as the last part.<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 24 / 29

Data, Moduli Spaces <strong>and</strong> Wild Speculation<br />

Question<br />

Does P(p ∤ # Jac C (F p )) = C p <br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 25 / 29

Data, Moduli Spaces <strong>and</strong> Wild Speculation<br />

Question<br />

Does P(p ∤ # Jac C (F p )) = C p <br />

Data<br />

- C hyperelliptic, p ≠ 2 – YES!<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 25 / 29

Data, Moduli Spaces <strong>and</strong> Wild Speculation<br />

Question<br />

Does P(p ∤ # Jac C (F p )) = C p <br />

Data<br />

- C hyperelliptic, p ≠ 2 – YES!<br />

- C plane curve, p ≠ 2 – YES!<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 25 / 29

Data, Moduli Spaces <strong>and</strong> Wild Speculation<br />

Question<br />

Does P(p ∤ # Jac C (F p )) = C p <br />

Data<br />

- C hyperelliptic, p ≠ 2 – YES!<br />

- C plane curve, p ≠ 2 – YES!<br />

- C plane curve, p = 2 –<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 25 / 29

Data, Moduli Spaces <strong>and</strong> Wild Speculation<br />

Question<br />

Does P(p ∤ # Jac C (F p )) = C p <br />

Data<br />

- C hyperelliptic, p ≠ 2 – YES!<br />

- C plane curve, p ≠ 2 – YES!<br />

- C plane curve, p = 2 – NO!!<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 25 / 29

C plane curve, p = 2<br />

Theorem (Cais, Ellenberg, ZB)<br />

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

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 26 / 29

C plane curve, p = 2<br />

Theorem (Cais, Ellenberg, ZB)<br />

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

Proof – theta characteristics.<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 26 / 29

a-number data<br />

Does<br />

∞∏ (<br />

P(a(Jac C (F p )) = 0) = 1 + p<br />

−i ) −1<br />

i=1<br />

∞∏ (<br />

= 1 − p<br />

−2i+1 ) <br />

i=1<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 27 / 29

a-number data<br />

Does<br />

P(a(Jac C (F p )) = 0) =<br />

Data<br />

- C hyperelliptic, p ≠ 2 –<br />

=<br />

∞∏ (<br />

1 + p<br />

−i ) −1<br />

i=1<br />

∞∏ (<br />

1 − p<br />

−2i+1 ) <br />

i=1<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 27 / 29

a-number data<br />

Does<br />

P(a(Jac C (F p )) = 0) =<br />

Data<br />

- C hyperelliptic, p ≠ 2 – not quite.<br />

=<br />

∞∏ (<br />

1 + p<br />

−i ) −1<br />

i=1<br />

∞∏ (<br />

1 − p<br />

−2i+1 ) <br />

i=1<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 27 / 29

a-number data<br />

Does<br />

P(a(Jac C (F p )) = 0) =<br />

Data<br />

- C hyperelliptic, p ≠ 2 – not quite.<br />

=<br />

∞∏ (<br />

1 + p<br />

−i ) −1<br />

i=1<br />

∞∏ (<br />

1 − p<br />

−2i+1 ) <br />

P(a(Jac C (F p )) = 0) = 1 − 3 −1 (p = 3)<br />

i=1<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 27 / 29

a-number data<br />

Does<br />

P(a(Jac C (F p )) = 0) =<br />

Data<br />

- C hyperelliptic, p ≠ 2 – not quite.<br />

=<br />

∞∏ (<br />

1 + p<br />

−i ) −1<br />

i=1<br />

∞∏ (<br />

1 − p<br />

−2i+1 ) <br />

P(a(Jac C (F p )) = 0) = 1 − 3 −1 (p = 3)<br />

i=1<br />

= (1 − 5 −1 )(1 − 5 −3 ) (p = 5)<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 27 / 29

a-number data<br />

Does<br />

P(a(Jac C (F p )) = 0) =<br />

Data<br />

- C hyperelliptic, p ≠ 2 – not quite.<br />

=<br />

∞∏ (<br />

1 + p<br />

−i ) −1<br />

i=1<br />

∞∏ (<br />

1 − p<br />

−2i+1 ) <br />

P(a(Jac C (F p )) = 0) = 1 − 3 −1 (p = 3)<br />

i=1<br />

= (1 − 5 −1 )(1 − 5 −3 ) (p = 5)<br />

= (1 − 7 −1 )(1 − 7 −3 )(1 − 7 −5 ) (p = 7)<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 27 / 29

<strong>Rational</strong> <strong>points</strong> on Moduli Spaces<br />

#Hg<br />

- P(a(Jac Cf (F p )) = 0) = lim ord(Fp)<br />

g→∞ #H g (F . p)<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 28 / 29

<strong>Rational</strong> <strong>points</strong> on Moduli Spaces<br />

#Hg<br />

- P(a(Jac Cf (F p )) = 0) = lim ord(Fp)<br />

g→∞ #H g (F . p)<br />

- One can access this through cohomology <strong>and</strong> the Weil conjectures.<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 28 / 29

<strong>Rational</strong> <strong>points</strong> on Moduli Spaces<br />

#Hg<br />

- P(a(Jac Cf (F p )) = 0) = lim ord(Fp)<br />

g→∞ #H g (F . p)<br />

- One can access this through cohomology <strong>and</strong> the Weil conjectures.<br />

- Our data suggests that Hg<br />

ord<br />

pulling back from H g .<br />

has cohomology that does not arise by<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 28 / 29

<strong>Rational</strong> <strong>points</strong> on Moduli Spaces<br />

#Hg<br />

- P(a(Jac Cf (F p )) = 0) = lim ord(Fp)<br />

g→∞ #H g (F . p)<br />

- One can access this through cohomology <strong>and</strong> the Weil conjectures.<br />

- Our data suggests that Hg<br />

ord<br />

pulling back from H g .<br />

has cohomology that does not arise by<br />

- P(a(Jac C (F p )) = 0) = lim g→∞<br />

#M ord<br />

g (F p)<br />

#M g (F p)<br />

= <br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 28 / 29

<strong>Rational</strong> <strong>points</strong> on Moduli Spaces<br />

#Hg<br />

- P(a(Jac Cf (F p )) = 0) = lim ord(Fp)<br />

g→∞ #H g (F . p)<br />

- One can access this through cohomology <strong>and</strong> the Weil conjectures.<br />

- Our data suggests that Hg<br />

ord<br />

pulling back from H g .<br />

has cohomology that does not arise by<br />

- P(a(Jac C (F p )) = 0) = lim g→∞<br />

#M ord<br />

g (F p)<br />

#M g (F p)<br />

= <br />

#A<br />

- P(a(A(F p )) = 0) = lim ord<br />

g (Fp)<br />

g→∞ #A g (F p)<br />

= <br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 28 / 29

Thank you<br />

Thank You!<br />

<strong>David</strong> <strong>Zureick</strong>-<strong>Brown</strong> (Emory University) R<strong>and</strong>om Dieudonné Modules November 13, 2012 29 / 29