The second moment of the size of the $2$-Selmer group of elliptic curves

TL;DR

This paper proves that the average of the square of the size of the 2-Selmer group for elliptic curves over rationals is bounded by 15, confirming a conjecture and advancing understanding of Selmer groups.

Contribution

It establishes an upper bound of 15 for the second moment of the 2-Selmer group size, confirming a conjecture by Poonen and Rains.

Findings

Second moment of 2-Selmer group size is at most 15.

Confirms conjecture of Poonen and Rains.

Advances understanding of elliptic curve Selmer groups.

Abstract

In this paper, we prove that when elliptic curves over $\mathbb{Q}$ are ordered by height, the second moment of the size of the $2$-Selmer group is at most $15$. This confirms a conjecture of Poonen and Rains.