Unrestricted pro-modularity at the prime two

3 papers

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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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 3

Unrestricted pro-modularity at the prime two

OpenAI · October 6, 2026

Every continuous, odd, absolutely irreducible two-dimensional 2-adic representation of \(G_{\mathbb Q}\) that is unramified outside finitely many finite primes occurs in the full completed Hecke algebra at some odd tame level. The level may contain auxiliary tame primes, and scalar and reducible residual representations are included. This is a completed-Hecke occurrence result, with no de Rham hypothesis; it does not assert classical modularity.

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

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@misc{OAI:Unrestricted-pro-modularity-at-the-prime-two-October-6-2026,
  author = {{OpenAI}},
  title = {{Unrestricted pro-modularity at the prime two}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026/two-adic-promodularity.pdf}{OAI:Unrestricted-pro-modularity-at-the-prime-two-October-6-2026}},
  year = {2026}
}

The Dimension of the Two-Adic Hecke Algebra at Odd Level

OpenAI · October 5, 2026

For every odd positive integer N, every irreducible component of the full two-adic Hecke algebra of level \(\Gamma_1(N)\) has Krull dimension four. This proves the p = 2 case of Emerton's dimension conjecture, including all residual components.

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@misc{OAI:The-Dimension-of-the-Two-Adic-Hecke-Algebra-at-Odd-Level-October-5-2026,
  author = {{OpenAI}},
  title = {{The Dimension of the Two-Adic Hecke Algebra at Odd Level}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Dimension-of-the-Two-Adic-Hecke-Algebra-at-Odd-Level-October-5-2026/two-adic-hecke.pdf}{OAI:The-Dimension-of-the-Two-Adic-Hecke-Algebra-at-Odd-Level-October-5-2026}},
  year = {2026}
}

Fontaine–Mazur modularity at the prime 2

OpenAI · October 6, 2026

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.

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

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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}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.