{"id":"openai-math/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank","pdf":"https://github.com/openai/math/blob/main/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf","url":"https://github.com/openai/math/tree/main/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026","tags":["number-theory","elliptic-curves","birch-swinnerton-dyer"],"score":0,"title":"Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank","domain":"Number theory","family":"002","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.NT","summary":"We prove the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full q -power Selmer group has corank zero or one at some prime q . The analytic and Mordell–Weil ranks equal that corank, and the Tate–Shafarevich group is finite. The formula includes all prime factors and requires no additional hypotheses on reduction, rational torsion, isogenies, complex multiplication, or residual Galois representations.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-10-07","reactions":{},"discussion":[{"url":"https://galoisrepresentations.org/2026/10/08/a-brief-digest-of-some-new-openai-results/#:~:text=My%20first%20impression%20is%20that%20these%20are%20technical%20improvements%20on%20results%20already%20in%20the%20literature","kind":"web","site":"galoisrepresentations.org","text":"My first impression is that these are technical improvements on results already in the literature, and that they represent no genuine progress towards the higher-rank version of BSD. At this point I don’t have so much interest in looking at them myself, but I would be happy to have a guest blog…","added":"2026-10-24","title":"A Brief Digest of Some New OpenAI Results | Persiflage","author":"Persiflage"}],"formalized":false,"family_title":"The full BSD formula from low Selmer corank"}