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

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.

19 pages

References in corpus (3)

• The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1
• The average number of elements in the 4-Selmer groups of elliptic curves is 7
• Invariant theory for the elliptic normal quintic, II. The covering map

Cited by in corpus (2)

• Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer
• Remarks on the Birch-Swinnerton-Dyer conjecture