The Geometric Distribution of Ranks and Selmer Groups of Elliptic - PowerPoint PPT Presentation

The Geometric Distribution of Ranks and Selmer Groups of Elliptic Curves over Function Fields Aaron Landesman (Stanford) Tony Feng (MIT) Eric Rains (Caltech) Special Session on Geometry and Topology in Arithmetic Madison, WI 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?

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 this and related conjectures hold for elliptic curves over F q ( t ) , in a q → ∞ limit.

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 E ( K ) / nE ( K ) H 1 ( Spec K , E [ n ]) H 1 ( Spec K , E )[ n ] 0 0 α ∏ v H 1 ( Spec K v , E v [ n ]) ∏ v H 1 ( Spec K v , E v )[ n ] F q E ( K v ) / nE v ( K v ) 0 ∏ v ∈ P 1 0. Definition The n -Selmer group of E is Sel n ( E ) : = ker α .

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 .

Conjecture on the average size of Selmer groups Conjecture (Bhargava–Shankar and Poonen–Rains) 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.

Conjectures on the distribution 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 Conjecture (Poonen–Rains) For ℓ a prime, as E ranges over elliptic curves over F q ( t ) , � � � � v 1 ℓ ∏ ∏ Prob ( dim F ℓ Sel ℓ ( E ) = v ) = . ℓ j − 1 1 + ℓ − j j ≥ 0 j = 1

The Poonen–Rains and BKLPR model Let O 2 r denote the orthogonal group associated to the quadratic form q 2 r = x 1 x 2 + x 3 x 4 + · · · + x 2 r − 1 x 2 r over F ℓ . Choose two random r -dimensional isotropic subspaces V and W . I.e., q 2 r | V = q 2 r | W = 0. Model (Poonen–Rains) The ℓ -Selmer group of an elliptic curve E is modeled as V ∩ W . So, Prob ( dim Sel ℓ ( E ) = α ) = lim r → ∞ Prob ( dim V ∩ W = α ) . The distribution of the rank of an elliptic curve is modeled as � if dim V ∩ W is even 0 rk ( E ) = 1 if dim V ∩ W is odd. Remark Bhargava, Kane, Lenstra, Poonen, and Rains have a similar but more sophisticated model applying for composite n .

Main result Definition Let ( rk BKLPR , Sel BKLPR ) denote the prediction of Bhargava, Kane, Lenstra, n Poonen, and Rains for the joint distribution of ranks and n -Selmer groups. Let ( rk , Sel n ) d F q denote the joint distribution of ranks and n -Selmer groups of elliptic curves of height ≤ d over F q ( t ) . Theorem (Feng-L-Rains) For any integer n ≥ 1 , we have   ( rk BKLPR , Sel BKLPR ( rk , Sel n ) d ) = lim  lim sup   n F q d → ∞  q → ∞ gcd ( q ,6 n )= 1    . ( rk , Sel n ) d = lim lim inf  F q q → ∞ d → ∞ gcd ( q ,6 n )= 1

Reversed limits Observe that we first take a large q limit, whereas BKLPR first takes a large height limit. Original Conjecture: Prob E : h ( E ) ≤ d ( Sel n ( E ) ≃ G ) = Prob (( rk BKLPR , Sel BKLPR ) = G ) . q → ∞ lim lim n # { E : h ( E ) ≤ d } d → ∞ Limits reversed: Prob E : h ( E ) ≤ d ( Sel n ( E ) ≃ G ) = Prob (( rk BKLPR , Sel BKLPR d → ∞ lim lim ) = G ) . n # { E : h ( E ) ≤ d } q → ∞ Remark Even though the limits are reversed, our results provide some of the first direct evidence for the connection between the arithmetic of elliptic curves and the complete conjectures of BKLPR.

Consequences Theorem (Feng-L-Rains)   ( rk BKLPR , Sel BKLPR ( rk , Sel n ) d ) = lim  lim sup   n F q d → ∞  q → ∞ gcd ( q ,6 n )= 1    . ( rk , Sel n ) d = lim lim inf  F q q → ∞ d → ∞ gcd ( q ,6 n )= 1 Corollary In the large q limit, the minimalist conjecture holds, meaning 50% of elliptic curves have rank 0 and 50% have rank 1 . Similarly, in the large q limit, the Poonen–Rains conjectures on average sizes, moments, and distributions of Selmer groups hold.

Theorem (Feng-L-Rains)   ( rk BKLPR , Sel BKLPR ( rk , Sel n ) d ) = lim  lim sup   n F q  d → ∞ q → ∞ gcd ( q ,6 n )= 1   ( rk , Sel n ) d  . = lim lim inf  F q q → ∞ d → ∞ gcd ( q ,6 n )= 1 Remark Similar results hold when working with quadratic twists of a fixed elliptic curve instead of all elliptic curves, as we just learned from Niudun Wang! Remark We write lim sup and lim inf because when d is even and n > 2, the limit does not exist. But the lim sup and lim inf tend to each other after further taking the height d → ∞ .

Proof overview The proof proceeds as follows: (1) Compare the actual distribution to a different model, which we call the “random kernel” model. (2) Show the random kernel model agrees with the BKLPR model in the large height limit. Model (Random kernel model) Here is the random kernel model for elliptic curves of height d over F q ( t ) : Let g ∈ O 12 d − 4 ( Z / n Z ) be a random element of the orthogonal group. Then, in the large q limit, Sel n ( E ) ≃ ker ( g − 1 ) . Further � 0 if g ∈ SO 12 d − 4 ( Z / n Z ) rk ( E ) = ∈ SO 12 d − 4 ( Z / n Z ) 1 if g /

Proof overview, continued The proof proceeds as follows: (1) Compare the actual distribution to a different model, which we call the “random kernel” model. (2) Show the random kernel model agrees with the BKLPR model in the large height limit. We check ( 2 ) directly for prime n , and deduce it for composite n by showing the two models satisfy the same Markov property. To check ( 1 ) , we relate the distribution of Selmer groups to ker ( g − id ) , for g a random element of an orthogonal group as follows: (A) Create a space parameterizing Selmer elements and covering the space of elliptic curves (B) Show the monodromy group of this cover is approximately an orthogonal group. (C) Use equidistribution of Frobenius to connect the actual distribution to our random kernel distribution.

Step (A): Create a space parameterizing Selmer elements 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 ) .

Step (B): Compute the monodromy group We have constructed a space Sel d n , k whose k points parameterize Selmer elements. Let W ◦ d k ⊂ W d k be the dense open parameterizing smooth Weierstrass models. Set up the fiber square Sel ◦ d Sel d n , k n , k π ◦ π W ◦ d W d k . k The resulting map π ◦ is finite ´ etale. Hence, we obtain a monodromy representation (or Galois representation) 1 ( W ◦ d ρ d n , k : π ´ et k ) → GL ( V d n , k ) . Theorem For n prime, there is a quadratic form Q d n on V d n , k so that, up to index 2 , im ρ d n , k = O ( Q d n ) .

Step (C): Connect the actual distribution to the random kernel model We have now constructed a monodromy representation. 1 ( W ◦ d ρ d n , k : π ´ et k ) → GL ( V d n , k ) . Theorem For n prime, there is a quadratic form Q d n on V d n , k so that, up to index 2 , im ρ d n , k = O ( Q d n ) . We can connect this to the random kernel model as follows: (I) The n -Selmer group of an elliptic curve [ E ] is realized as the F q points of a fiber of π : Sel d n , k → W d k . But, F q points are the same as points fixed by Frobenius, i.e., ker ( ρ d n , k ( Frob ) − id ) . (II) Equidistribution of ρ d n , k ( Frob ) in the monodromy group realizes the Selmer group as ker ( g − id ) , for g a random element of the monodromy group im ρ d n , k . This is our random kernel model.