The geometric average size of Selmer groups over function fields Aaron Landesman Stanford University Number Theory, Arithmetic Geometry, and Computation II Baltimore, MD Slides available at http://www.web.stanford.edu/~aaronlan/slides/
Ranks of elliptic curves Theorem (Mordell-Weil) Let E be an elliptic curve over a global field K (such as Q or F q ( t ) ). Then the group of K-rational points E ( K ) is a finitely generated abelian group. For E an elliptic curve over K , write E ( K ) ≃ Z r ⊕ T for T a finite group. Then, r is the rank of E . Question What is the average rank of an elliptic curve? Aaron Landesman The geometric average size of Selmer groups over function fields 2 / 14
Motivation Conjecture (Minimalist Conjecture) The average rank of elliptic curves is 1 / 2. Moreover, • 50% of curves have rank 0, • 50% have rank 1, • 0% have rank more than 1. Goal Explain why 0% of elliptic curves have rank more than 1, in a certain sense. Aaron Landesman The geometric average size of Selmer groups over function fields 3 / 14
Definition of Selmer group Let K = F q ( t ) , and let v index the closed points of P 1 F q . For E an elliptic curve over K , the multiplication by n exact sequence × n E [ n ] 0 E E 0 induces the sequences on ´ etale cohomology β H 1 ( Spec K , E [ n ]) H 1 ( Spec K , E )[ n ] 0 E ( K ) / nE ( K ) 0 α ∏ v H 1 ( Spec K v , E v [ n ]) ∏ v H 1 ( Spec K v , E v )[ n ] 0 ∏ v ∈ P 1 F q E ( K v ) / nE v ( K v ) 0. Definition The n -Selmer group of E is Sel n ( E ) : = ker α . Aaron Landesman The geometric average size of Selmer groups over function fields 4 / 14
Selmer group and rank β H 1 ( Spec K , E [ n ]) H 1 ( Spec K , E )[ n ] 0 E ( K ) / nE ( K ) 0 α ∏ v H 1 ( Spec K v , E v [ n ]) ∏ v H 1 ( Spec K v , E v )[ n ] 0 F q E ( K v ) / nE v ( K v ) 0. ∏ v ∈ P 1 Lemma There is an injection E ( K ) / nE ( K ) → Sel n ( E ) . Proof. E ( K ) / nE ( K ) = ker β ⊂ ker α = Sel n ( E ) . Corollary The Z / n rank of Sel n ( E ) is an upper bound for the rank of E. Aaron Landesman The geometric average size of Selmer groups over function fields 5 / 14
Average size of Selmer groups Say E / F q ( t ) is in minimal Weierstrass form given by y 2 z = x 3 + A ( s , t ) xz 2 + B ( s , t ) z 3 , (so char F q > 3,) where there exists d so that A ( s , t ) and B ( s , t ) are homogeneous polynomials in F q [ s , t ] of degrees 4 d and 6 d . The height of E is h ( E ) : = d . Definition The average size of the n -Selmer group of height up to d is Average ≤ d ( # Sel n / F q ( t )) : = ∑ E / F q ( t ) , h ( E ) ≤ d # Sel n ( E ) # { E / F q ( t ) : h ( E ) ≤ d } , where the sum runs over isomorphism classes of elliptic curves E / F q ( t ) , having h ( E ) ≤ d . Aaron Landesman The geometric average size of Selmer groups over function fields 6 / 14
Conjecture on the average size of Selmer groups Conjecture (Bhargava–Shankar and Poonen–Rains) When all elliptic curves are ordered by height, d → ∞ Average ≤ d ( # Sel n / F q ( t )) = ∑ q → ∞ lim lim s . s | n Remark • An analogous statement over Q (without a limit in q ) was shown for n = 2, 3, 4, 5 by Bhargava and Shankar. • The upper bound was shown for n = 3 over F q ( t ) by de Jong. • This was shown for n = 2 more generally over function fields by Ho, Le Hung, and Ngo. Aaron Landesman The geometric average size of Selmer groups over function fields 7 / 14
Application of conjecture to ranks Assuming the conjecture, 100% of elliptic curves have rank at most 1: Corollary (Assuming conjecture) Let P ≤ d denote the proportion of elliptic curves of rank ≥ 2 over F q ( t ) of q height up to d. If the conjecture were true, d → ∞ P ≤ d q → ∞ lim lim = 0. q Proof. Take n prime. Since n rk E ≤ # Sel n ( E ) , in the limit we have = n 2 Average ( δ rk ( E ) ≥ 2 ) ≤ Average ( n rk E ) n 2 P ≤ d q ≤ Average ( # Sel n ( E )) = ∑ s = n + 1. s | n Since, n 2 P ≤ d ≤ n + 1, taking n large shows P ≤ d → 0. q q Aaron Landesman The geometric average size of Selmer groups over function fields 8 / 14
Main result We can try to approach the conjecture by reversing the limits. ∑ E / F q , h ( E ) ≤ d # Sel n ( E ) = ∑ Conjecture: lim q → ∞ lim s . # { E : h ( E ) ≤ d } d → ∞ s | n ∑ E / F q , h ( E ) ≤ d # Sel n ( E ) = ∑ Limits reversed: lim d → ∞ lim s . # { E : h ( E ) ≤ d } q → ∞ s | n Theorem (L.) For n ≥ 1 and d ≥ 2 , Average ≤ d ( # Sel n / F q ( t )) = ∑ lim s . q → ∞ s | n gcd ( q ,2 n )= 1 Aaron Landesman The geometric average size of Selmer groups over function fields 9 / 14
Application to ranks Theorem (L.) For n ≥ 1 and d ≥ 2 , Average ≤ d ( # Sel n / F q ( t )) = ∑ lim s . q → ∞ s | n gcd ( q ,2 n )= 1 Analogously to the corollary to the conjecture, 100% of elliptic curves of height up to d have rank at most 1 in the large q limit: Corollary If P ≤ d denotes the proportion of elliptic curve of rank ≥ 2 over F q ( t ) of q height up to d, for d ≥ 2 , P ≤ d lim = 0. q q → ∞ gcd ( q ,2 n )= 1 Aaron Landesman The geometric average size of Selmer groups over function fields 10 / 14
Three heuristics for the average size of Selmer groups Question Why is the average size of the n -Selmer group ∑ s | n s ? • In the known cases over Q , the proof connects the average size to Tamagawa number τ ( PGL s ) = s , and the average size is ∑ s | n τ ( PGL s ) • We show, via a monodromy computation, that the average size is the number of orbits of a certain orthogonal group. If n is prime, these orbits are the 0 orbit and the n level sets of the associated quadratic form. • The average size is the number of balanced rank s projective bundles over P 1 for s | n , which are of the form Proj P 1 Sym • ( O ⊕ a ⊕ O ( − 1 ) ⊕ s − a ) for 1 ≤ a ≤ s . Altogether, there are ∑ s | n s such bundles as s ranges over the divisors of n . Aaron Landesman The geometric average size of Selmer groups over function fields 11 / 14
Higher moments of Selmer groups Let Average ≤ d (( # Sel n ) m / F q ( t )) denote the average size of Sel n ( E ) m as E varies through elliptic curves over F q ( t ) of height ≤ d . Conjecture (Poonen-Rains) For ℓ prime and m ≥ 1, d → ∞ Average ≤ d (( # Sel ℓ ) m / F q ( t )) = ( 1 + ℓ )( 1 + ℓ 2 ) · · · ( 1 + ℓ m ) . q → ∞ lim lim Using a more involved monodromy computation, we can prove: Theorem (Feng-L) Let ℓ be prime and m ≥ 1 . For d ≥ max ( 2, m + 4 6 ) , Average ≤ d (( # Sel ℓ ) m / F q ( t )) = ( 1 + ℓ )( 1 + ℓ 2 ) · · · ( 1 + ℓ m ) . lim q → ∞ gcd ( q ,2 n )= 1 Aaron Landesman The geometric average size of Selmer groups over function fields 12 / 14
Quadratic twists Definition For E a fixed elliptic curve over F q ( t ) defined by y 2 z = x 3 + A ( s , t ) xz 2 + B ( s , t ) z 3 , one can define the quadratic twist family of degree d as those elliptic curves of the form f ( s , t ) y 2 z = x 3 + A ( s , t ) xz 2 + B ( s , t ) z 3 , for f ( s , t ) ∈ k [ s , t ] varying over square-free homogeneous polynomial of degree d . Remark (Average sizes, with a twist!) Sun Woo Park and Niudun Wang proved that the average size of n -Selmer groups in certain quadratic twist families is ∑ s | n s , using a similar method. Aaron Landesman The geometric average size of Selmer groups over function fields 13 / 14
Proof overview Theorem (L) For n ≥ 1 and d ≥ 2 , Average ≤ d ( # Sel n / F q ( t )) = ∑ lim s . q → ∞ s | n gcd ( q ,2 n )= 1 Proof overview: (1) Construct a space Sel d n , k parameterizing n -Selmer elements of elliptic curves of height d over k . (2) By Lang-Weil, the average size of the n -Selmer group is the number of components of Sel d n , k (3) Compute the number of components of Sel d n , k by viewing it as a finite cover of the moduli of height d elliptic curves, and computing the monodromy. Aaron Landesman The geometric average size of Selmer groups over function fields 14 / 14
Proof sketch For k a finite field, construct a space Sel d n , k parameterizing pairs ( E , X ) , where E is an elliptic curve over k ( t ) and X is an n -Selmer element of E . Letting W d k denote a parameter space for Weierstrass equations of elliptic curves E / k ( t ) of height d . There is a projection map π : Sel d n , k → W d k ( E , X ) �→ [ E ] . The key property is π − 1 ([ E ])( k ) = Sel n ( E ) . The total number of Selmer elements over varying elliptic curves is Sel d n , k ( k ) , so we are reduced to computing # Sel d n , k ( k ′ ) # W d k ( k ′ ) for large finite extensions k ′ of k . Aaron Landesman The geometric average size of Selmer groups over function fields 14 / 14
Proof sketch, continued We want to compute # Sel d n , k ( k ′ ) k ( k ′ ) . # W d Theorem (Lang-Weil) For X a finite type space over F p with r geometrically irreducible components, lim q → ∞ X ( F q ) = rq dim X + O ( q dim X − 1 / 2 ) . So, # Sel d k ( k ′ ) = #components of Sel d n , k ( k ′ ) n , k # W d #components of W d k = #components of Sel d n , k 1 = #components of Sel d n , k . Aaron Landesman The geometric average size of Selmer groups over function fields 14 / 14
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