The average number of elements in the 4-Selmer groups of elliptic curves is 7

TL;DR

This paper proves that the average size of 4-Selmer groups of elliptic curves over rationals is 7, revealing insights into the distribution of Selmer groups and the Tate-Shafarevich group structure.

Contribution

It establishes the average size of 4-Selmer groups for all elliptic curves over ield, a new result in the arithmetic statistics of elliptic curves.

Findings

Average 4-Selmer group size is 7

At least 20% of 2-Selmer elements do not lift to 4-Selmer elements

A positive proportion of elliptic curves have nontrivial 2-torsion in Sha

Abstract

We prove that when all elliptic curves over $\mathbb{Q}$ are ordered by height, the average size of their 4-Selmer groups is equal to 7. As a consequence, we show that a positive proportion (in fact, at least one fifth) of all 2-Selmer elements of elliptic curves, when ordered by height, do not lift to 4-Selmer elements, and thus correspond to nontrivial 2-torsion elements in the associated Tate--Shafarevich groups.