A positive proportion of elliptic curves over $\mathbb{Q}$ have rank one

TL;DR

This paper proves that a positive proportion of elliptic curves over the rationals have rank one, implying that the average rank and average analytic rank are both strictly positive.

Contribution

It establishes that, when ordered by naive height, a positive proportion of elliptic curves have both algebraic and analytic rank one, a significant advance in understanding their distribution.

Findings

A positive proportion of elliptic curves have rank one.

The average rank of elliptic curves is strictly positive.

The average analytic rank of elliptic curves is strictly positive.

Abstract

We prove that, when all elliptic curves over $\mathbb{Q}$ are ordered by naive height, a positive proportion have both algebraic and analytic rank one. It follows that the average rank and the average analytic rank of elliptic curves are both strictly positive.