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

Manjul Bhargava, Christopher Skinner

Published 2014-01-01Version 1

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.