The geometric distribution of Selmer groups of elliptic curves over function fields

Abstract

Fix a positive integer n and a finite field {mathbb {F}}_q. We study the joint distribution of the rank {{,mathrm{rk},}}(E), the n-Selmer group text {Sel}_n(E), and the n-torsion in the Tate–Shafarevich group as E varies over elliptic curves of fixed height d ge 2 over {mathbb {F}}_q(t). We compute this joint distribution in the large q limit. We also show that the “large q, then large height” limit of this distribution agrees with the one predicted by Bhargava–Kane–Lenstra–Poonen–Rains.