{"id":"openai-math/Fontaine-Mazur-modularity-at-the-prime-2","pdf":"https://github.com/openai/math/blob/main/preprints/Fontaine-Mazur-modularity-at-the-prime-2-October-6-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Fontaine-Mazur-modularity-at-the-prime-2-October-6-2026","tags":["number-theory","galois-representations","modularity"],"score":0,"title":"Fontaine–Mazur modularity at the prime 2","domain":"Number theory","family":"010","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.NT","summary":"We prove that every continuous, irreducible, odd two-dimensional 2-adic representation of $`G_{\\mathbb Q}`$, unramified outside finitely many primes and de Rham at 2 with distinct Hodge–Tate weights, is modular up to Tate twist. This resolves the odd, regular two-dimensional Fontaine–Mazur conjecture over ℚ at 2, including all residual representations.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-10-06","reactions":{},"discussion":[{"url":"https://galoisrepresentations.org/2026/10/08/a-brief-digest-of-some-new-openai-results/#:~:text=It%20is%20very%20clean%2C%20which%20is%20useful%20for%20applications.","kind":"web","site":"galoisrepresentations.org","text":"It is very clean, which is useful for applications. That said, upon closer inspection this paper may turn out to have some really clever new trick, but I can’t see what that is on a superficial reading. I would actually say I’m surprised that this is a problem OpenAI would have tried to solve,…","added":"2026-10-24","title":"A Brief Digest of Some New OpenAI Results | Persiflage","author":"Persiflage"}],"formalized":false,"family_title":"Unrestricted pro-modularity at the prime two"}