Random Dieudonné Modules and the Cohen-Lenstra Heuristics
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1 Random Dieudonné Modules and the Cohen-Lenstra Heuristics David Zureick-Brown Bryden Cais Jordan Ellenberg Emory University Slides available at Arithmetic of abelian varieties in families Lausanne, Switzerland November 13, 2012
2 Basic Question How often does p divide h( D)? David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, / 29
3 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, / 29
4 Guess: Random Integer? P(p h( D)) = P(p D) = 1 p??? David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, / 29
5 Data (Buell 76) P(p h( D)) 1 p + 1 p 2 1 p (p odd ) p7 = 1 (1 1p ) i i 1 = /3 (p = 3) = /5 (p = 5) P(Cl( D) 3 = Z/9Z) P(Cl( D) 3 = (Z/3Z) 2 ) David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, / 29
6 Random finite abelian groups Idea P(p h( D)) = P(p #G) =??? David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, / 29
7 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, / 29
8 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, / 29
9 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, / 29
10 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, / 29
11 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, / 29
12 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, / 29
13 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, / 29
14 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, / 29
15 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, / 29
16 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, / 29
17 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, / 29
18 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, / 29
19 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, / 29
20 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, / 29
21 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, / 29
22 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, / 29
23 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, / 29
24 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, / 29
25 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, / 29
26 Random Tate-modules F GL 2g (Z l ) (w/ Haar measure) David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, / 29
27 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, / 29
28 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, / 29
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, / 29
30 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, / 29
31 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, / 29
32 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, / 29
33 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, / 29
34 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, / 29
35 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, / 29
36 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, / 29
37 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, / 29
38 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, / 29
39 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, / 29
40 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, / 29
41 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, / 29
42 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, / 29
43 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, / 29
44 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, / 29
45 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, / 29
46 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, / 29
47 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, / 29
48 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, / 29
49 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, / 29
50 Proofs Part (i) Mod pqp D has a natural probability measure. David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, / 29
51 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, / 29
52 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, / 29
53 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, / 29
54 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, / 29
55 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, / 29
56 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, / 29
57 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, / 29
58 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, / 29
59 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, / 29
60 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, / 29
61 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, / 29
62 Proofs Part (iii) P(r(M) = g s) = complicated but explicit expression. David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, / 29
63 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, / 29
64 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, / 29
65 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, / 29
66 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, / 29
67 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, / 29
68 Proofs Part (iv) 1 st moment is 2: Avg (#G(F p )[p]) = 2 David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, / 29
69 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, / 29
70 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, / 29
71 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, / 29
72 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, / 29
73 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, / 29
74 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, / 29
75 Proofs Part (v) P ( p # coker(f Id) F (M F p)) = Cp. David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, / 29
76 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, / 29
77 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, / 29
78 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, / 29
79 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, / 29
80 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, / 29
81 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, / 29
82 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, / 29
83 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, / 29
84 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, / 29
85 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, / 29
86 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, / 29
87 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) = (p = 3) i=1 David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, / 29
88 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) = (p = 3) i=1 = (1 5 1 )(1 5 3 ) (p = 5) David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, / 29
89 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) = (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, / 29
90 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, / 29
91 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, / 29
92 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, / 29
93 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, / 29
94 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, / 29
95 Thank you Thank You! David Zureick-Brown (Emory University) Random Dieudonné Modules November 13, / 29
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