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

Abstract

Fix a positive integer n and a finite field $${\mathbb {F}}_q$$ F q . We study the joint distribution of the rank $${{\,\mathrm{rk}\,}}(E)$$ rk ( E ) , the n-Selmer group $$\text {Sel}_n(E)$$ Sel n ( E ) , and the n-torsion in the Tate–Shafarevich group "Equation missing" as E varies over elliptic curves of fixed height $$d \ge 2$$ d ≥ 2 over $${\mathbb {F}}_q(t)$$ 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.