The Birch and Swinnerton-Dyer conjecture implies Goldfeld's conjecture

2025/03/22 by Smith, Alexander · 5 citations
#11G05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2503.17619

Abstract

Given an elliptic curve E/Q, we show that 50% of the quadratic twists of E have 2^∞-Selmer corank 0 and 50% have 2^∞-Selmer corank 1. As one consequence, we prove that the Birch and Swinnerton-Dyer conjecture implies Goldfeld's conjecture. Previously, this result was known by work of the author for elliptic curves over Q satisfying certain technical conditions. As part of this work, we determine the distribution of 2-Selmer ranks in the quadratic twist family of E. In the cases where this distribution was not already known, it is distinct from the model for distributions of 2-Selmer groups constructed by Poonen and Rains.

Cited by

• Selmer ranks under quadratic twists satisfying the Heegner hypothesis
• Iwasawa Theory of Elliptic Curves in Quadratic Twist Families
• Infinitely many pairs of non-isomorphic elliptic curves sharing the same BSD invariants
• A Topological Perspective on the Birch and Swinnerton Dyer Conjectures
• Elliptic curves of rank one over number fields

Related

• Generalized Birch lemma and the 2-part of the Birch and Swinnerton-Dyer conjecture for certain elliptic curves
• 2^∞-Selmer groups, 2^∞-class groups, and Goldfeld's conjecture
• On the Distribution of 2-Selmer Ranks within Quadratic Twist Families of Elliptic Curves with Partial Rational Two-Torsion
• On Twists of A Family of Elliptic Curves and Their L-Function
• Iwasawa theory and ranks of elliptic curves in quadratic twist families