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

We prove Goldfeld's analytic density conjecture: for every elliptic curve E over 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.