oai-math-revised beta

Browse › Number theory › 010 Unrestricted pro-modularity at the prime two

Unrestricted pro-modularity at the prime two

OpenAI · October 4, 2026

Earlier edition

This is an earlier edition (October 4, 2026). Current edition: Unrestricted pro-modularity at the prime two, October 6, 2026.

Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion

Edit NOTES.md · Existing issues · Source on GitHub

If the PDF does not display here, open it directly: Open PDF.

Notes

No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:Unrestricted-pro-modularity-at-the-prime-two-October-4-2026,
  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-4-2026/two-adic-promodularity.pdf}{OAI:Unrestricted-pro-modularity-at-the-prime-two-October-4-2026}},
  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.

Other papers in this family: