David Zureick-Brown - Rational points and algebraic cycles
Random Dieudonné Modules and the Cohen-Lenstra
Heuristics
David Zureick-Brown
Bryden Cais
Jordan Ellenberg
Emory University
Slides available at http://www.mathcs.emory.edu/~dzb/slides/
Arithmetic of abelian varieties in families
Lausanne, Switzerland
November 13, 2012
Basic Question
How often does p divide h(−D)
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 2 / 29
Basic Question
What is
P(p | h(−D)) = lim
X →∞
#{0 ≤ D ≤ X s.t. p | h(−D)}
#{0 ≤ D ≤ X }
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 3 / 29
Guess: Random Integer
P(p | h(−D)) = P(p | D) = 1 p
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 4 / 29
Data (Buell ’76)
P(p | h(−D)) ≈ 1 p + 1 p 2 − 1 p 5 − 1 + · · · (p odd )
p7 = 1 − ∏ (1 − 1 )
p i i≥1
= 0.43 . . . ≠ 1/3 (p = 3)
= 0.23 . . . ≠ 1/5 (p = 5)
P(Cl(−D) 3
∼ = Z/9Z) ≈ 0.070
P(Cl(−D) 3
∼ = (Z/3Z) 2 ) ≈ 0.0097
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 5 / 29
Random finite abelian groups
Idea
P(p | h(−D)) = P(p | #G) =
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 6 / 29
Random finite abelian groups
Idea
P(p | h(−D)) = P(p | #G) =
Let G p be the set of isomorphism classes of finite abelian groups of
p-power order.
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 6 / 29
Random finite abelian groups
Idea
P(p | h(−D)) = P(p | #G) =
Let G p be the set of isomorphism classes of finite abelian groups of
p-power order.
Theorem (Cohen, Lenstra)
(i)
∑ 1
# Aut G = ∏ G∈G p i
(
1 − 1 p i ) −1
= C −1
p
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 6 / 29
Random finite abelian groups
Idea
P(p | h(−D)) = P(p | #G) =
Let G p be the set of isomorphism classes of finite abelian groups of
p-power order.
Theorem (Cohen, Lenstra)
(i)
∑ 1
# Aut G = ∏ G∈G p i
(ii) G ↦→
(
1 − 1 p i ) −1
= C −1
p
C p
# Aut G is a probability distribution on G p
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 6 / 29
Random finite abelian groups
Idea
P(p | h(−D)) = P(p | #G) =
Let G p be the set of isomorphism classes of finite abelian groups of
p-power order.
Theorem (Cohen, Lenstra)
(i)
∑ 1
# Aut G = ∏ G∈G p i
(ii) G ↦→
(
1 − 1 p i ) −1
= C −1
p
C p
# Aut G is a probability distribution on G p
(iii) Avg (#G[p]) = Avg ( p rp(G)) = 2
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 6 / 29
Cohen and Lenstra’s conjecture
Let f : G p → Z be a function.
Definition
Avg f = ∑
G∈G p
C p
# Aut G · f (G)
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 7 / 29
Cohen and Lenstra’s conjecture
Let f : G p → Z be a function.
Definition
Avg f = ∑
G∈G p
C p
# Aut G · f (G)
Avg Cl f =
∑
0≤D≤X f (Cl(−D) p)
∑
0≤D≤X 1
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 7 / 29
Cohen and Lenstra’s conjecture
Let f : G p → Z be a function.
Definition
Avg f = ∑
G∈G p
C p
# Aut G · f (G)
Avg Cl f =
∑
0≤D≤X f (Cl(−D) p)
∑
0≤D≤X 1
Conjecture (Cohen, Lenstra)
(i) Avg Cl f = Avg f
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 7 / 29
Cohen and Lenstra’s conjecture
Let f : G p → Z be a function.
Definition
Avg f = ∑
G∈G p
C p
# Aut G · f (G)
Avg Cl f =
∑
0≤D≤X f (Cl(−D) p)
∑
0≤D≤X 1
Conjecture (Cohen, Lenstra)
(i) Avg Cl f = Avg f
(ii) Avg (# Cl(−D)[p]) = 2
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 7 / 29
Cohen and Lenstra’s conjecture
Let f : G p → Z be a function.
Definition
Avg f = ∑
G∈G p
C p
# Aut G · f (G)
Avg Cl f =
∑
0≤D≤X f (Cl(−D) p)
∑
0≤D≤X 1
Conjecture (Cohen, Lenstra)
(i) Avg Cl f = Avg f
(ii) Avg (# Cl(−D)[p]) 2 = 2 + p
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 7 / 29
Cohen and Lenstra’s conjecture
Let f : G p → Z be a function.
Definition
Avg f = ∑
G∈G p
C p
# Aut G · f (G)
Avg Cl f =
∑
0≤D≤X f (Cl(−D) p)
∑
0≤D≤X 1
Conjecture (Cohen, Lenstra)
(i) Avg Cl f = Avg f
(ii) Avg (# Cl(−D)[p]) 2 = 2 + p
(iii) P(Cl(−D) p
∼ = G) =
C p
# Aut G .
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 7 / 29
Progress
Davenport-Heilbronn – Avg Cl(−D)[3] = 2
Bhargava – Avg Cl(K)[2] = 3 (K cubic)
Bhargava – counts quartic dihedral extensions
Kohnen-Ono – N p ∤h (X ) ≫ x 2
1
log x
Heath-Brown – N p|h (X ) ≫ x 10
9
log x
Byeon – N Clp ∼ =(Z/gZ)
2(X ) ≫ x 1 g
log x
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 8 / 29
Cohen-Lenstra over F q (t), l ≠ p
Cl(−D) = Pic(Spec O K )
vs
Pic(C)
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 9 / 29
Cohen-Lenstra over F q (t), l ≠ p
Cl(−D) = Pic(Spec O K )
vs
Pic(C) deg
−→ Z → 0
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 9 / 29
Cohen-Lenstra over F q (t), l ≠ p
Cl(−D) = Pic(Spec O K )
vs
0 → Pic 0 (C) → Pic(C) deg
−→ Z → 0
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 9 / 29
Basic Question over F q (t), l ≠ p
Fix G ∈ G l .
What is
P(Pic 0 (C) l
∼ = G)
(Limit is taken as deg f → ∞, where C : y 2 = f (x).)
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 10 / 29
Main Tool over F q (t) – Tate Module
Aut T l (Jac C ) ∼ = Z 2g
l
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 11 / 29
Main Tool over F q (t) – Tate Module
Gal Fq
→ Aut T l (Jac C ) ∼ = Z 2g
l
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 11 / 29
Main Tool over F q (t) – Tate Module
Frob ∈ Gal Fq
→ Aut T l (Jac C ) ∼ = Z 2g
l
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 11 / 29
Main Tool over F q (t) – Tate Module
- Frob ∈ Gal Fq → Aut T l (Jac C ) ∼ = Z 2g
l
- coker (Frob − Id) ∼ = Jac C (F q ) l = Pic 0 (C)
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 11 / 29
Random Tate-modules
F ∈ GL 2g (Z l ) (w/ Haar measure)
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 12 / 29
Random Tate-modules
F ∈ GL 2g (Z l ) (w/ Haar measure)
Theorem (Friedman, Washington)
P(coker F − I ∼ = L) =
C l
# Aut L
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 12 / 29
Random Tate-modules
F ∈ GL 2g (Z l ) (w/ Haar measure)
Theorem (Friedman, Washington)
Conjecture
P(coker F − I ∼ = L) =
P(Pic 0 (C) ∼ = L) =
C l
# Aut L
C l
# Aut L
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 12 / 29
Progress
In the limit (w/ upper and lower densities):
Achter – conjectures are true for GSp 2g instead of GL 2g .
Ellenberg-Venkatesh – conjectures are true if l ∤ q − 1.
Garton – explicit conjectures for GSp 2g , l | q − 1.
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 13 / 29
Cohen-Lenstra over F p (t), l = p
Basic question – what is
P(p | # Jac C (F p ))
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 14 / 29
Cohen-Lenstra over F p (t), l = p
T l (Jac C ) ∼ = Z r l , 0 ≤ r ≤ g
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 15 / 29
Cohen-Lenstra over F p (t), l = p
Definition
The p-rank of Jac C is the integer r.
T l (Jac C ) ∼ = Z r l , 0 ≤ r ≤ g
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 15 / 29
Cohen-Lenstra over F p (t), l = p
Definition
The p-rank of Jac C is the integer r.
Complication
T l (Jac C ) ∼ = Z r l , 0 ≤ r ≤ g
As C varies, r varies. Need to know the distribution of p-ranks, or find a
better algebraic gadget than T l (Jac C ).
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 15 / 29
Dieudonné Modules
Definition
(i) D = Z q [F , V ]/(FV = VF = p, Fz = z σ F , Vz = z σ−1 V ).
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 16 / 29
Dieudonné Modules
Definition
(i) D = Z q [F , V ]/(FV = VF = p, Fz = z σ F , Vz = z σ−1 V ).
(ii) A Dieudonné module is a D-module which is finite and free as a Z q
module.
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 16 / 29
Dieudonné Modules
Definition
(i) D = Z q [F , V ]/(FV = VF = p, Fz = z σ F , Vz = z σ−1 V ).
(ii) A Dieudonné module is a D-module which is finite and free as a Z q
module.
Jac C
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 16 / 29
Dieudonné Modules
Definition
(i) D = Z q [F , V ]/(FV = VF = p, Fz = z σ F , Vz = z σ−1 V ).
(ii) A Dieudonné module is a D-module which is finite and free as a Z q
module.
Jac C
M = H 1 cris (Jac C, Z p )
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 16 / 29
Dieudonné Modules
Definition
(i) D = Z q [F , V ]/(FV = VF = p, Fz = z σ F , Vz = z σ−1 V ).
(ii) A Dieudonné module is a D-module which is finite and free as a Z q
module.
Jac C
M = H 1 cris (Jac C, Z p )
{Jac C [p n ]} n
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 16 / 29
Dieudonné Modules
Definition
(i) D = Z q [F , V ]/(FV = VF = p, Fz = z σ F , Vz = z σ−1 V ).
(ii) A Dieudonné module is a D-module which is finite and free as a Z q
module.
Jac C
M = H 1 cris (Jac C, Z p )
H 1 dR (Jac C, F p )
{Jac C [p n ]} n
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 16 / 29
Dieudonné Modules
Definition
(i) D = Z q [F , V ]/(FV = VF = p, Fz = z σ F , Vz = z σ−1 V ).
(ii) A Dieudonné module is a D-module which is finite and free as a Z q
module.
Jac C
M = H 1 cris (Jac C, Z p )
H 1 dR (Jac C, F p )
{Jac C [p n ]} n
V −1 : df ↦→ “d(f p )”
p
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 16 / 29
Invariants via Dieudonné Modules
Invariants
(i) p-rank(Jac C ) = dim F ∞ (M ⊗ F p ).
(ii) a(Jac C ) = dim Hom(α p , Jac C [p]) = dim (ker V ∩ ker F ).
(iii) Jac C (F p ) p = coker(F − Id)| F ∞ (M⊗F p).
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 17 / 29
Principally quasi polarized Dieudoneé modules
Definition
A principally quasi polarized Dieudoneé module a Dieudoneé module M
together with a non-degenerate symplectic pairing 〈 , 〉 such that for all
x, y ∈ M,
〈Fx, y〉 = σ〈x, Vy〉.
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 18 / 29
Main Theorem
Theorem (Cais, Ellenberg, ZB)
(i) Mod pqp D has a natural probability measure.
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 19 / 29
Main Theorem
Theorem (Cais, Ellenberg, ZB)
(i) Mod pqp D has a natural probability measure.
(Push forward along Sp 2g (Z p ) 2 → Sp 2g (Z p ) · F 0 · Sp 2g (Z p ))
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 19 / 29
Main Theorem
Theorem (Cais, Ellenberg, ZB)
(i) Mod pqp D has a natural probability measure.
(Push forward along Sp 2g (Z p ) 2 → Sp 2g (Z p ) · F 0 · Sp 2g (Z p ))
(ii) P(a(M) = s) = p −(s+1 2 ) ·
∞∏ (
1 + p
−i ) −1
·
i=1
s∏ (
1 − p
−i ) −1
.
i=1
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 19 / 29
Main Theorem
Theorem (Cais, Ellenberg, ZB)
(i) Mod pqp D has a natural probability measure.
(Push forward along Sp 2g (Z p ) 2 → Sp 2g (Z p ) · F 0 · Sp 2g (Z p ))
(ii) P(a(M) = s) = p −(s+1 2 ) ·
∞∏ (
1 + p
−i ) −1
·
i=1
s∏ (
1 − p
−i ) −1
.
(iii) P(r(M) = g − s) = complicated but explicit expression.
i=1
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 19 / 29
Main Theorem
Theorem (Cais, Ellenberg, ZB)
(i) Mod pqp D has a natural probability measure.
(Push forward along Sp 2g (Z p ) 2 → Sp 2g (Z p ) · F 0 · Sp 2g (Z p ))
(ii) P(a(M) = s) = p −(s+1 2 ) ·
∞∏ (
1 + p
−i ) −1
·
i=1
s∏ (
1 − p
−i ) −1
.
(iii) P(r(M) = g − s) = complicated but explicit expression.
(iii’) P(r(M) = g − 2) = (p −2 + p −3 ) ·
i=1
∞∏ (
1 + p
−i ) −1
i=1
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 19 / 29
Main Theorem
Theorem (Cais, Ellenberg, ZB)
(i) Mod pqp D has a natural probability measure.
(Push forward along Sp 2g (Z p ) 2 → Sp 2g (Z p ) · F 0 · Sp 2g (Z p ))
(ii) P(a(M) = s) = p −(s+1 2 ) ·
∞∏ (
1 + p
−i ) −1
·
i=1
s∏ (
1 − p
−i ) −1
.
(iii) P(r(M) = g − s) = complicated but explicit expression.
(iii’) P(r(M) = g − 2) = (p −2 + p −3 ) ·
(iv) 1 st moment is 2.
i=1
∞∏ (
1 + p
−i ) −1
i=1
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 19 / 29
Main Theorem
Theorem (Cais, Ellenberg, ZB)
(i) Mod pqp D has a natural probability measure.
(Push forward along Sp 2g (Z p ) 2 → Sp 2g (Z p ) · F 0 · Sp 2g (Z p ))
(ii) P(a(M) = s) = p −(s+1 2 ) ·
∞∏ (
1 + p
−i ) −1
·
i=1
s∏ (
1 − p
−i ) −1
.
(iii) P(r(M) = g − s) = complicated but explicit expression.
(iii’) P(r(M) = g − 2) = (p −2 + p −3 ) ·
(iv) 1 st moment is 2.
i=1
∞∏ (
1 + p
−i ) −1
i=1
(v) P ( p ∤ # coker(F − Id)| F ∞ (M⊗F p))
= Cp .
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 19 / 29
Proofs
Part (i)
Mod pqp D has a natural probability measure.
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 20 / 29
Proofs
Part (i)
Mod pqp D has a natural probability measure.
1 (D, 〈 , 〉, F , V ) s.t., FV = VF = p and 〈F (−) , −〉 = σ〈− , V (−)〉.
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 20 / 29
Proofs
Part (i)
Mod pqp D has a natural probability measure.
1 (D, 〈 , 〉, F , V ) s.t., FV = VF = p and 〈F (−) , −〉 = σ〈− , V (−)〉.
⎡ ⎤ ⎡ ⎤
2 D = Z 2g ⎢
q , 〈 , 〉 = ⎣ 0 I
−I 0
⎥
⎦, F 0 =
⎢
⎣ pI 0
0 I
⎥
⎦, V 0 = pF −1 .
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 20 / 29
Proofs
Part (i)
Mod pqp D has a natural probability measure.
1 (D, 〈 , 〉, F , V ) s.t., FV = VF = p and 〈F (−) , −〉 = σ〈− , V (−)〉.
⎡ ⎤ ⎡ ⎤
2 D = Z 2g ⎢
q , 〈 , 〉 = ⎣ 0 I
−I 0
Proposition
⎥
⎦, F 0 =
⎢
⎣ pI 0
0 I
⎥
⎦, V 0 = pF −1 .
The double coset space Sp 2g (Z p ) · F 0 · Sp 2g (Z p ) contains all pqp
Dieudoneé modules.
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 20 / 29
Proofs
Part (i)
Mod pqp D has a natural probability measure.
1 (D, 〈 , 〉, F , V ) s.t., FV = VF = p and 〈F (−) , −〉 = σ〈− , V (−)〉.
⎡ ⎤ ⎡ ⎤
2 D = Z 2g ⎢
q , 〈 , 〉 = ⎣ 0 I
−I 0
Proposition
⎥
⎦, F 0 =
⎢
⎣ pI 0
0 I
⎥
⎦, V 0 = pF −1 .
The double coset space Sp 2g (Z p ) · F 0 · Sp 2g (Z p ) contains all pqp
Dieudoneé modules.
Proof: Witt’s theorem – Sp 2g acts transitively on symplecto-bases.
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 20 / 29
Proofs
Part (i)
Mod pqp D has a natural probability measure.
1 (D, 〈 , 〉, F , V ) s.t., FV = VF = p and 〈F (−) , −〉 = σ〈− , V (−)〉.
⎡ ⎤ ⎡ ⎤
2 D = Z 2g ⎢
q , 〈 , 〉 = ⎣ 0 I
−I 0
Proposition
⎥
⎦, F 0 =
⎢
⎣ pI 0
0 I
⎥
⎦, V 0 = pF −1 .
The double coset space Sp 2g (Z p ) · F 0 · Sp 2g (Z p ) contains all pqp
Dieudoneé modules.
Proof: Witt’s theorem – Sp 2g acts transitively on symplecto-bases.
Note: F ∉ Sp 2g (Z p ), but rather the subset of GSp 2g (Z p ) of multiplier p g
matricies.
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 20 / 29
Proofs
Part (ii)
P(a(M) = s) = p −(s+1 2 ) ·
∞∏ (
1 + p
−i ) −1
·
i=1
s∏ (
1 − p
−i ) −1
.
i=1
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 21 / 29
Proofs
Part (ii)
P(a(M) = s) = p −(s+1 2 ) ·
∞∏ (
1 + p
−i ) −1
·
i=1
s∏ (
1 − p
−i ) −1
.
1 Duality implies that W 1 := ker(F ⊗ F p ) and W 2 := ker(V ⊗ F p ) are
maximal isotropics.
i=1
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 21 / 29
Proofs
Part (ii)
P(a(M) = s) = p −(s+1 2 ) ·
∞∏ (
1 + p
−i ) −1
·
i=1
s∏ (
1 − p
−i ) −1
.
1 Duality implies that W 1 := ker(F ⊗ F p ) and W 2 := ker(V ⊗ F p ) are
maximal isotropics.
2 a(M) = dim (W 1 ∩ W 2 )
i=1
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 21 / 29
Proofs
Part (ii)
P(a(M) = s) = p −(s+1 2 ) ·
∞∏ (
1 + p
−i ) −1
·
i=1
s∏ (
1 − p
−i ) −1
.
1 Duality implies that W 1 := ker(F ⊗ F p ) and W 2 := ker(V ⊗ F p ) are
maximal isotropics.
2 a(M) = dim (W 1 ∩ W 2 )
i=1
3 Argue that W 1 and W 2 are randomly distributed.
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 21 / 29
Proofs
Part (ii)
P(a(M) = s) = p −(s+1 2 ) ·
∞∏ (
1 + p
−i ) −1
·
i=1
s∏ (
1 − p
−i ) −1
.
1 Duality implies that W 1 := ker(F ⊗ F p ) and W 2 := ker(V ⊗ F p ) are
maximal isotropics.
2 a(M) = dim (W 1 ∩ W 2 )
i=1
3 Argue that W 1 and W 2 are randomly distributed.
4 This expression is the probability that two random maximal isotropics
intersect with dimension s.
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 21 / 29
Proofs
Part (ii)
P(a(M) = s) = p −(s+1 2 ) ·
∞∏ (
1 + p
−i ) −1
·
i=1
s∏ (
1 − p
−i ) −1
.
1 Duality implies that W 1 := ker(F ⊗ F p ) and W 2 := ker(V ⊗ F p ) are
maximal isotropics.
2 a(M) = dim (W 1 ∩ W 2 )
i=1
3 Argue that W 1 and W 2 are randomly distributed.
4 This expression is the probability that two random maximal isotropics
intersect with dimension s.
5 Compute this with Witt’s theorem (Sp 2g acts transitively on pairs of
maximal isotropics whose intersection has dimension s), and compute
explicitly the size of the stabilizers.
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 21 / 29
Proofs
Part (iii)
P(r(M) = g − s) = complicated but explicit expression.
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 22 / 29
Proofs
Part (iii)
P(r(M) = g − s) = complicated but explicit expression.
1 Recall: r(M) = dim F ∞ (M) = rank(F ⊗ F p ) g .
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 22 / 29
Proofs
Part (iii)
P(r(M) = g − s) = complicated but explicit expression.
1 Recall: r(M) = dim F ∞ (M) = rank(F ⊗ F p ) g .
2 (Prüfer, Crabb, others) The number of nilpotent N ∈ M n (F q ) is
q n(n−1) . Able to modify Crabb’s argument:
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 22 / 29
Proofs
Part (iii)
P(r(M) = g − s) = complicated but explicit expression.
1 Recall: r(M) = dim F ∞ (M) = rank(F ⊗ F p ) g .
2 (Prüfer, Crabb, others) The number of nilpotent N ∈ M n (F q ) is
q n(n−1) . Able to modify Crabb’s argument:
1 Given N nilpotent, get a flag V i := N i (V ).
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 22 / 29
Proofs
Part (iii)
P(r(M) = g − s) = complicated but explicit expression.
1 Recall: r(M) = dim F ∞ (M) = rank(F ⊗ F p ) g .
2 (Prüfer, Crabb, others) The number of nilpotent N ∈ M n (F q ) is
q n(n−1) . Able to modify Crabb’s argument:
1 Given N nilpotent, get a flag V i := N i (V ).
2 There is a unique basis {y 1 , . . . , y g } such that N(y g ) = 0 and
V i = 〈N i (y mi +1), . . . , N(y g−1 )〉 (where m i = g − dim V i−1 )
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 22 / 29
Proofs
Part (iii)
P(r(M) = g − s) = complicated but explicit expression.
1 Recall: r(M) = dim F ∞ (M) = rank(F ⊗ F p ) g .
2 (Prüfer, Crabb, others) The number of nilpotent N ∈ M n (F q ) is
q n(n−1) . Able to modify Crabb’s argument:
1 Given N nilpotent, get a flag V i := N i (V ).
2 There is a unique basis {y 1 , . . . , y g } such that N(y g ) = 0 and
V i = 〈N i (y mi +1), . . . , N(y g−1 )〉 (where m i = g − dim V i−1 )
3 The map N ↦→ (N(y 1 ), . . . , N(y g−1 )) ∈ V n−1 is bijective.
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 22 / 29
Proofs
Part (iv)
1 st moment is 2: Avg (#G(F p )[p]) = 2
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 23 / 29
Proofs
Part (iv)
1 st moment is 2: Avg (#G(F p )[p]) = 2
1 First fix the p-corank.
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 23 / 29
Proofs
Part (iv)
1 st moment is 2: Avg (#G(F p )[p]) = 2
1 First fix the p-corank.
1 Associated p-divisible group decomposes as
G = G m × G et × G ll .
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 23 / 29
Proofs
Part (iv)
1 st moment is 2: Avg (#G(F p )[p]) = 2
1 First fix the p-corank.
1 Associated p-divisible group decomposes as
G = G m × G et × G ll .
2 Fixing the p-corank fixes the dimension of G ll
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 23 / 29
Proofs
Part (iv)
1 st moment is 2: Avg (#G(F p )[p]) = 2
1 First fix the p-corank.
1 Associated p-divisible group decomposes as
G = G m × G et × G ll .
2 Fixing the p-corank fixes the dimension of G ll
2 (Show that G random ⇒ G et random.)
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 23 / 29
Proofs
Part (iv)
1 st moment is 2: Avg (#G(F p )[p]) = 2
1 First fix the p-corank.
1 Associated p-divisible group decomposes as
G = G m × G et × G ll .
2 Fixing the p-corank fixes the dimension of G ll
2 (Show that G random ⇒ G et random.)
3 G(F p ) = G et (F p ) = coker(F | M et − Id).
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 23 / 29
Proofs
Part (iv)
1 st moment is 2: Avg (#G(F p )[p]) = 2
1 First fix the p-corank.
1 Associated p-divisible group decomposes as
G = G m × G et × G ll .
2 Fixing the p-corank fixes the dimension of G ll
2 (Show that G random ⇒ G et random.)
3 G(F p ) = G et (F p ) = coker(F | M et − Id).
4 F | M et is random in GL g (Z p ).
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 23 / 29
Proofs
Part (v)
P ( p ∤ # coker(F − Id)| F ∞ (M⊗F p))
= Cp .
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 24 / 29
Proofs
Part (v)
P ( p ∤ # coker(F − Id)| F ∞ (M⊗F p))
= Cp .
Basically the same proof as the last part.
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 24 / 29
Data, Moduli Spaces and Wild Speculation
Question
Does P(p ∤ # Jac C (F p )) = C p
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 25 / 29
Data, Moduli Spaces and Wild Speculation
Question
Does P(p ∤ # Jac C (F p )) = C p
Data
- C hyperelliptic, p ≠ 2 – YES!
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 25 / 29
Data, Moduli Spaces and Wild Speculation
Question
Does P(p ∤ # Jac C (F p )) = C p
Data
- C hyperelliptic, p ≠ 2 – YES!
- C plane curve, p ≠ 2 – YES!
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 25 / 29
Data, Moduli Spaces and Wild Speculation
Question
Does P(p ∤ # Jac C (F p )) = C p
Data
- C hyperelliptic, p ≠ 2 – YES!
- C plane curve, p ≠ 2 – YES!
- C plane curve, p = 2 –
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 25 / 29
Data, Moduli Spaces and Wild Speculation
Question
Does P(p ∤ # Jac C (F p )) = C p
Data
- C hyperelliptic, p ≠ 2 – YES!
- C plane curve, p ≠ 2 – YES!
- C plane curve, p = 2 – NO!!
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 25 / 29
C plane curve, p = 2
Theorem (Cais, Ellenberg, ZB)
P(2 ∤ # Jac C (F 2 )) = 0 for plane curves of odd degree.
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 26 / 29
C plane curve, p = 2
Theorem (Cais, Ellenberg, ZB)
P(2 ∤ # Jac C (F 2 )) = 0 for plane curves of odd degree.
Proof – theta characteristics.
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 26 / 29
a-number data
Does
∞∏ (
P(a(Jac C (F p )) = 0) = 1 + p
−i ) −1
i=1
∞∏ (
= 1 − p
−2i+1 )
i=1
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 27 / 29
a-number data
Does
P(a(Jac C (F p )) = 0) =
Data
- C hyperelliptic, p ≠ 2 –
=
∞∏ (
1 + p
−i ) −1
i=1
∞∏ (
1 − p
−2i+1 )
i=1
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 27 / 29
a-number data
Does
P(a(Jac C (F p )) = 0) =
Data
- C hyperelliptic, p ≠ 2 – not quite.
=
∞∏ (
1 + p
−i ) −1
i=1
∞∏ (
1 − p
−2i+1 )
i=1
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 27 / 29
a-number data
Does
P(a(Jac C (F p )) = 0) =
Data
- C hyperelliptic, p ≠ 2 – not quite.
=
∞∏ (
1 + p
−i ) −1
i=1
∞∏ (
1 − p
−2i+1 )
P(a(Jac C (F p )) = 0) = 1 − 3 −1 (p = 3)
i=1
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 27 / 29
a-number data
Does
P(a(Jac C (F p )) = 0) =
Data
- C hyperelliptic, p ≠ 2 – not quite.
=
∞∏ (
1 + p
−i ) −1
i=1
∞∏ (
1 − p
−2i+1 )
P(a(Jac C (F p )) = 0) = 1 − 3 −1 (p = 3)
i=1
= (1 − 5 −1 )(1 − 5 −3 ) (p = 5)
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 27 / 29
a-number data
Does
P(a(Jac C (F p )) = 0) =
Data
- C hyperelliptic, p ≠ 2 – not quite.
=
∞∏ (
1 + p
−i ) −1
i=1
∞∏ (
1 − p
−2i+1 )
P(a(Jac C (F p )) = 0) = 1 − 3 −1 (p = 3)
i=1
= (1 − 5 −1 )(1 − 5 −3 ) (p = 5)
= (1 − 7 −1 )(1 − 7 −3 )(1 − 7 −5 ) (p = 7)
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 27 / 29
Rational points on Moduli Spaces
#Hg
- P(a(Jac Cf (F p )) = 0) = lim ord(Fp)
g→∞ #H g (F . p)
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 28 / 29
Rational points on Moduli Spaces
#Hg
- P(a(Jac Cf (F p )) = 0) = lim ord(Fp)
g→∞ #H g (F . p)
- One can access this through cohomology and the Weil conjectures.
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 28 / 29
Rational points on Moduli Spaces
#Hg
- P(a(Jac Cf (F p )) = 0) = lim ord(Fp)
g→∞ #H g (F . p)
- One can access this through cohomology and the Weil conjectures.
- Our data suggests that Hg
ord
pulling back from H g .
has cohomology that does not arise by
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 28 / 29
Rational points on Moduli Spaces
#Hg
- P(a(Jac Cf (F p )) = 0) = lim ord(Fp)
g→∞ #H g (F . p)
- One can access this through cohomology and the Weil conjectures.
- Our data suggests that Hg
ord
pulling back from H g .
has cohomology that does not arise by
- P(a(Jac C (F p )) = 0) = lim g→∞
#M ord
g (F p)
#M g (F p)
=
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 28 / 29
Rational points on Moduli Spaces
#Hg
- P(a(Jac Cf (F p )) = 0) = lim ord(Fp)
g→∞ #H g (F . p)
- One can access this through cohomology and the Weil conjectures.
- Our data suggests that Hg
ord
pulling back from H g .
has cohomology that does not arise by
- P(a(Jac C (F p )) = 0) = lim g→∞
#M ord
g (F p)
#M g (F p)
=
#A
- P(a(A(F p )) = 0) = lim ord
g (Fp)
g→∞ #A g (F p)
=
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 28 / 29
Thank you
Thank You!
David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, 2012 29 / 29
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