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Fontaine–Mazur modularity at the prime 2

OpenAI · September 23, 2026

Earlier edition

This is an earlier edition (September 23, 2026). Current edition: Fontaine–Mazur modularity at the prime 2, October 6, 2026.

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@misc{OAI:Fontaine-Mazur-modularity-at-the-prime-2-September-23-2026,
  author = {{OpenAI}},
  title = {{Fontaine--Mazur modularity at the prime 2}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Fontaine-Mazur-modularity-at-the-prime-2-September-23-2026/paper.pdf}{OAI:Fontaine-Mazur-modularity-at-the-prime-2-September-23-2026}},
  year = {2026}
}

Family 010: Unrestricted pro-modularity at the prime two

Every continuous odd absolutely irreducible two-dimensional 2-adic representation of $G_{\mathbb Q}$ unramified outside finitely many primes occurs in a completed Hecke algebra at some odd tame level. Also proves classical modularity up to Tate twist for irreducible odd representations with these finiteness conditions that are de Rham at 2 with distinct Hodge–Tate weights, resolving the dyadic Fontaine–Mazur case without residual restrictions.

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