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

TL;DR

This paper investigates the distribution of Selmer groups, ranks, and Tate-Shafarevich groups of elliptic curves over function fields, confirming predictions from a probabilistic model in the large finite field limit.

Contribution

It computes the joint distribution of these invariants for elliptic curves over function fields and verifies the agreement with the Bhargava-Kane-Lenstra-Poonen-Rains heuristic in a specific limit.

Findings

Distribution matches the predicted probabilistic model

Large finite field limit confirms theoretical predictions

Provides explicit computations for 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 of $E$, the $n$-Selmer group of $E$, and the $n$-torsion in the Tate-Shafarevich group of $E$ as $E$ varies over elliptic curves of fixed height $d \geq 2$ 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.