Goldfeld's analytic density conjecture and the 2-converse for elliptic curves

Abstract — v1

We prove Goldfeld's analytic density conjecture: for every elliptic curve $E$ over $\mathbb Q$, the quadratic twists of $E$ with analytic rank zero and one each have density $1/2$ among signed squarefree twist parameters ordered by absolute value. We also prove the low-corank $2$-converse: if the $2^\infty$-Selmer corank of $E$ is zero or one, then it equals the analytic and Mordell–Weil ranks, and the Tate–Shafarevich group is finite.