Result 010, Number theory

Unrestricted pro-modularity at the prime two

Every continuous odd absolutely irreducible two-dimensional 2-adic representation of GQG_{\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.

Proof

The bigger picture

Why it matters

The manuscripts describe a broad bridge between symmetries of algebraic numbers and modular forms, highly structured functions in number theory. At the prime two, the claimed bridge includes even cases with scalar or reducible reductions modulo two.

What changes?

The main manuscript reports that every continuous, odd, absolutely irreducible two-dimensional 2-adic representation of the rational Galois group, unramified outside finitely many primes, occurs in a full completed Hecke algebra at some odd tame level. Such representations encode arithmetic symmetries; "odd" means complex conjugation has determinant minus one. The algebra collects modular-form operators and their 2-adic limits. The level may include auxiliary primes. No de Rham hypothesis is required, but occurrence in this algebra does not assert classical modularity.

What does that help mathematicians do?

A companion manuscript reports a stronger conclusion for continuous, irreducible, odd two-dimensional 2-adic representations with the same finite-ramification condition: classical modularity up to Tate twist, provided they are de Rham at two and have distinct Hodge-Tate weights. These extra local conditions let researchers identify the symmetry data with those of a classical modular form, after a standard adjustment called a Tate twist. This conclusion imposes no restrictions on the residual representation, the data obtained by reduction modulo two.

Are there practical applications?

The immediate value is foundational: a more uniform description of arithmetic symmetries through modular forms. Another companion manuscript reports that, for every odd positive integer N, every irreducible component of the full two-adic Hecke algebra at level Gamma-one of N has Krull dimension four, including all residual components. This supplies a uniform geometric constraint for studying these algebras, rather than a demonstrated technological application.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

Unrestricted pro-modularity at the prime two

October 4, 2026 29 pages

Every continuous, odd, absolutely irreducible two-dimensional 2-adic representation of GQG_{\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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

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

October 5, 2026 12 pages

For every odd positive integer N, every irreducible component of the full two-adic Hecke algebra of level Γ1(N)\Gamma_1(N) has Krull dimension four. This proves the p = 2 case of Emerton's dimension conjecture, including all residual components.

PDF LaTeX source
Cite (BibTeX) 
@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

September 23, 2026 72 pages

We prove that every continuous, irreducible, odd two-dimensional 2-adic representation of GQG_{\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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.