David Zureick-Brown - Rational points and algebraic cycles
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