The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1

TL;DR

This paper proves that the average rank of elliptic curves over the rationals is less than 1, based on the average size of their 5-Selmer groups and distribution of root numbers, with implications for the distribution of ranks.

Contribution

It establishes the average size of the 5-Selmer group as 6 and provides new bounds on the average rank of elliptic curves over ield, improving understanding of their rank distribution.

Findings

Average 5-Selmer group size is 6.

At least 80% of elliptic curves have rank 0 or 1.

At least 20% of elliptic curves have rank 0.

Abstract

In this article, we prove that the average rank of elliptic curves over $\mathbb{Q}$, when ordered by height, is less than $1$ (in fact, less than $.885$). As a consequence of our methods, we also prove that at least four fifths of all elliptic curves over $\mathbb{Q}$ have rank either 0 or 1; furthermore, at least one fifth of all elliptic curves in fact have rank 0. The primary ingredient in the proofs of these theorems is a determination of the average size of the $5$-Selmer group of elliptic curves over $\mathbb{Q}$; we prove that this average size is $6$. Another key ingredient is a new lower bound on the equidistribution of root numbers of elliptic curves; we prove that there is a family of elliptic curves over $\mathbb{Q}$ having density at least $55\%$ for which the root number is equidistributed.