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arXiv · 1401.0233

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

Manjul Bhargava·Christopher Skinner
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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.

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Manjul Bhargava, Christopher Skinner. 2014-01-01. A positive proportion of elliptic curves over $\mathbb{Q}$ have rank one. https://arxiv.org/abs/1401.0233

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