The Selmer converse for elliptic curves at every prime

We prove the Selmer converse in coranks zero and one for every elliptic curve over ℚ and every prime p: if the full p-power Selmer group has ℤ${}_p$-corank $r\in \{0,1\}$, then the analytic and Mordell–Weil ranks both equal r, and the entire Tate–Shafarevich group is finite. As an application at the additive prime 3, we prove that for every prime $\ell \equiv 4,7,8(\mathrm{mod}9)$, the cubic $X^3+Y^3=\ell Z^3$ has analytic and Mordell–Weil rank one and finite Tate–Shafarevich group. In particular, every such ℓ is a sum of two rational cubes.