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

Abstract

Given an elliptic curve E/Q, we show that 50% of the quadratic twists of E have $2^{\infty}$-Selmer corank 0 and 50% have $2^{\infty}$-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.