Random Dieudonn´ e Modules and the Cohen-Lenstra Heuristics David Zureick-Brown Bryden Cais Jordan Ellenberg Emory University Slides available at http://www.mathcs.emory.edu/~dzb/slides/ Arithmetic of abelian varieties in families Lausanne, Switzerland November 13, 2012
Basic Question How often does p divide h ( − D )? David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 2 / 29
Basic Question What is # { 0 ≤ D ≤ X s.t. p | h ( − D ) } P ( p | h ( − D )) = lim ? # { 0 ≤ D ≤ X } X →∞ David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 3 / 29
Guess : Random Integer? P ( p | h ( − D )) = P ( p | D ) = 1 p ??? David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 4 / 29
Data (Buell ’76) P ( p | h ( − D )) ≈ 1 p + 1 p 2 − 1 p 5 − 1 p 7 + · · · ( p odd ) � 1 − 1 � � = 1 − p i i ≥ 1 = 0 . 43 . . . � = 1 / 3 ( p = 3) = 0 . 23 . . . � = 1 / 5 ( p = 5) P (Cl( − D ) 3 ∼ = Z / 9 Z ) ≈ 0 . 070 P (Cl( − D ) 3 ∼ = ( Z / 3 Z ) 2 ) ≈ 0 . 0097 David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 5 / 29
Random finite abelian groups Idea P ( p | h ( − D )) = P ( p | # G ) = ??? David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 6 / 29
Random finite abelian groups Idea P ( p | h ( − D )) = P ( p | # G ) = ??? Let G p be the set of isomorphism classes of finite abelian groups of p -power order. David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 6 / 29
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) � − 1 1 � 1 − 1 � � = C − 1 (i) # Aut G = p p i G ∈ G p i David Zureick-Brown (Emory University) Random Dieudonn´ e 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) � − 1 1 � 1 − 1 � � = C − 1 (i) # Aut G = p p i G ∈ G p i C p (ii) G �→ # Aut G is a probability distribution on G p David Zureick-Brown (Emory University) Random Dieudonn´ e 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) � − 1 1 � 1 − 1 � � = C − 1 (i) # Aut G = p p i G ∈ G p i C p (ii) G �→ # Aut G is a probability distribution on G p p r p ( G ) � � (iii) Avg (# G [ p ]) = Avg = 2 David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 6 / 29
Cohen and Lenstra’s conjecture Let f : G p → Z be a function. Definition C p � Avg f = # Aut G · f ( G ) G ∈ G p David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 7 / 29
Cohen and Lenstra’s conjecture Let f : G p → Z be a function. Definition C p � Avg f = # Aut G · f ( G ) G ∈ G p � 0 ≤ D ≤ X f (Cl( − D ) p ) Avg Cl f = � 0 ≤ D ≤ X 1 David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 7 / 29
Cohen and Lenstra’s conjecture Let f : G p → Z be a function. Definition C p � Avg f = # Aut G · f ( G ) G ∈ G p � 0 ≤ D ≤ X f (Cl( − D ) p ) Avg Cl f = � 0 ≤ D ≤ X 1 Conjecture (Cohen, Lenstra) (i) Avg Cl f = Avg f David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 7 / 29
Cohen and Lenstra’s conjecture Let f : G p → Z be a function. Definition C p � Avg f = # Aut G · f ( G ) G ∈ G p � 0 ≤ D ≤ X f (Cl( − D ) p ) Avg Cl f = � 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 Dieudonn´ e Modules November 13, 2012 7 / 29
Cohen and Lenstra’s conjecture Let f : G p → Z be a function. Definition C p � Avg f = # Aut G · f ( G ) G ∈ G p � 0 ≤ D ≤ X f (Cl( − D ) p ) Avg Cl f = � 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 Dieudonn´ e Modules November 13, 2012 7 / 29
Cohen and Lenstra’s conjecture Let f : G p → Z be a function. Definition C p � Avg f = # Aut G · f ( G ) G ∈ G p � 0 ≤ D ≤ X f (Cl( − D ) p ) Avg Cl f = � 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 ∼ C p = G ) = # Aut G . David Zureick-Brown (Emory University) Random Dieudonn´ e 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 1 x 2 Kohnen-Ono – N p ∤ h ( X ) ≫ log x 9 N p | h ( X ) ≫ x 10 Heath-Brown – log x 1 x g Byeon – N Cl p ∼ =( Z / g Z ) 2 ( X ) ≫ log x David Zureick-Brown (Emory University) Random Dieudonn´ e 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 Dieudonn´ e Modules November 13, 2012 9 / 29
Cohen-Lenstra over F q ( t ), ℓ � = p Cl( − D ) = Pic(Spec O K ) vs deg Pic( C ) − − → Z → 0 David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 9 / 29
Cohen-Lenstra over F q ( t ), ℓ � = p Cl( − D ) = Pic(Spec O K ) vs deg 0 → Pic 0 ( C ) → Pic( C ) − − → Z → 0 David Zureick-Brown (Emory University) Random Dieudonn´ e 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 Dieudonn´ e Modules November 13, 2012 10 / 29
Main Tool over F q ( t ) – Tate Module Aut T ℓ (Jac C ) ∼ = Z 2 g ℓ David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 11 / 29
Main Tool over F q ( t ) – Tate Module Gal F q → Aut T ℓ (Jac C ) ∼ = Z 2 g ℓ David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 11 / 29
Main Tool over F q ( t ) – Tate Module Frob ∈ Gal F q → Aut T ℓ (Jac C ) ∼ = Z 2 g ℓ David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 11 / 29
Main Tool over F q ( t ) – Tate Module - Frob ∈ Gal F q → Aut T ℓ (Jac C ) ∼ = Z 2 g ℓ - coker (Frob − Id) ∼ = Jac C ( F q ) ℓ = Pic 0 ( C ) David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 11 / 29
Random Tate-modules F ∈ GL 2 g ( Z ℓ ) (w/ Haar measure) David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 12 / 29
Random Tate-modules F ∈ GL 2 g ( Z ℓ ) (w/ Haar measure) Theorem (Friedman, Washington) C ℓ P (coker F − I ∼ = L ) = # Aut L David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 12 / 29
Random Tate-modules F ∈ GL 2 g ( Z ℓ ) (w/ Haar measure) Theorem (Friedman, Washington) C ℓ P (coker F − I ∼ = L ) = # Aut L Conjecture C ℓ P (Pic 0 ( C ) ∼ = L ) = # Aut L David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 12 / 29
Progress In the limit (w/ upper and lower densities): Achter – conjectures are true for GSp 2 g instead of GL 2 g . Ellenberg-Venkatesh – conjectures are true if ℓ ∤ q − 1. Garton – explicit conjectures for GSp 2 g , ℓ | q − 1. David Zureick-Brown (Emory University) Random Dieudonn´ e 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 Dieudonn´ e 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 Dieudonn´ e Modules November 13, 2012 15 / 29
Cohen-Lenstra over F p ( t ), ℓ = p T ℓ (Jac C ) ∼ = Z r ℓ , 0 ≤ r ≤ g Definition The p - rank of Jac C is the integer r . David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 15 / 29
Cohen-Lenstra over F p ( t ), ℓ = p T ℓ (Jac C ) ∼ = Z r ℓ , 0 ≤ r ≤ g Definition The p - rank of Jac C is the integer r . Complication As C varies, r varies. Need to know the distribution of p -ranks, or find a better algebraic gadget than T ℓ (Jac C ). David Zureick-Brown (Emory University) Random Dieudonn´ e Modules November 13, 2012 15 / 29
Dieudonn´ e 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 Dieudonn´ e Modules November 13, 2012 16 / 29
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