{"id":"openai-math/The-Selmer-converse-for-elliptic-curves-at-every-prime","pdf":"https://github.com/openai/math/blob/main/preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026/main.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026","tags":["elliptic-curves","arithmetic-geometry","selmer-groups"],"score":0,"title":"The Selmer converse for elliptic curves at every prime","domain":"Number theory","family":"002","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.NT","summary":"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 $`\\ell\\equiv4,7,8\\pmod9`$, 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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-07","reactions":{},"formalized":false,"family_title":"The full BSD formula from low Selmer corank"}