Remark: 002-02. The Selmer converse for elliptic curves at every prime

Authors: OpenAI. Family 002.

Title: The Selmer converse for elliptic curves at every prime.

Abstract: 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 ℤp-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 ${}ell{}equiv4,7,8{}pmod9$, 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.

This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.