OpenAI · 002-02 · 2026-10-07 · Number theory
The Selmer converse for elliptic curves at every prime
preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026
Family 002. The full Birch–Swinnerton-Dyer formula in Selmer coranks zero and one.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Proves the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over whose full -power Selmer group has corank zero or one for some prime , including finiteness of the Tate–Shafarevich group. With result 006, this gives full BSD for a density-one set of quadratic twists of every elliptic curve over .
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 , 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 , the cubic has analytic and Mordell–Weil rank one and finite Tate–Shafarevich group. In particular, every such ℓ is a sum of two rational cubes.
Preprint source and Lean scope
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No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.