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Fontaine–Mazur modularity at the prime 2

OpenAI · October 6, 2026

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Abstract

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.

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Comments and discussion

  • Persiflage (galoisrepresentations.org), A Brief Digest of Some New OpenAI Results (blog post, 2026-10-08). Item #2 gives an explicitly superficial assessment of this paper, suggesting the main difficulties at p=2 are technical, relating it to recent work of Lue Pan, calling it "very clean, which is useful for applications" and noting its use in the release's Hilbert's tenth problem argument. In the comments, Chris Xu writes: "#2 feels like folklore that people didn't care to write down."

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Cite

@misc{OAI:Fontaine-Mazur-modularity-at-the-prime-2-October-6-2026,
  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-October-6-2026/paper.pdf}{OAI:Fontaine-Mazur-modularity-at-the-prime-2-October-6-2026}},
  year = {2026}
}

Versions

This version updates citations to revised manuscripts in this collection; see the previous version.

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.

Other papers in this family: