A majority of elliptic curves over $\mathbb{Q}$ satisfy the Birch and Swinnerton-Dyer conjecture

Abstract

We prove that a majority (in fact, $>66\%$) of all elliptic curves over $\mathbb{Q}$, when ordered by height, satisfy the Birch and Swinnerton-Dyer rank conjecture.

arXiv commentSome further details, clarifications, and references added; 17 pages

References in corpus (2)

• The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1
• Indivisibility of Heegner points in the multiplicative case

Cited by in corpus (10)

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• Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer
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• A positive proportion of cubic curves over Q admit linear determinantal representations
• Congruences between Heegner points and quadratic twists of elliptic curves
• Behaviour of the order of Tate-Shafarevich groups for the quadratic twists of elliptic curves
• Heegner points at Eisenstein primes and twists of elliptic curves
• Elliptic curves with exceptionally large analytic order of the Tate-Shafarevich groups
• Factorization of measures and applications to the weak Goldfeld conjecture