Quadratic twist families of elliptic curves with unusual $2^{\infty}$-Selmer groups

Given any elliptic curve $E$ over the rationals, 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 a result, we show that Goldfeld's conjecture follows from the Birch and Swinnerton--Dyer conjecture.

Unlike the distribution of $2^{\infty}$-Selmer coranks, the distribution of $2^{\infty}$-Selmer groups in the twist family of $E$ can depend on the choice of $E$. For most quadratic twist families, the distribution agrees with the heuristic of Bhargava, Kane, Lenstra, Poonen, and Rains; but there are five other types of distributions that can appear. In this talk, we will focus on twist families that have more unusual Selmer group distributions, and we highlight the complications that arise in these cases when applying our machinery for finding the distribution of Selmer group