{"id":"openai-math/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion","pdf":"https://github.com/openai/math/blob/main/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026","tags":["elliptic-curves","selmer-groups","arithmetic-geometry"],"score":0,"title":"A pointwise 2-converse for elliptic curves with rational two-torsion","domain":"Number theory","family":"004","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.NT","summary":"We prove a pointwise 2-converse for elliptic curves over $`\\mathbf Q`$ with nonzero rational two-torsion: if the $`2^\\infty`$-Selmer corank is zero or one, then the analytic rank and Mordell–Weil rank equal that corank, and the Shafarevich–Tate group is finite. The result allows arbitrary reduction at 2.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-07","reactions":{},"formalized":false,"family_title":"Hilbert’s tenth problem over ℚ"}