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The Fontaine–Mazur conjecture at the prime 2

From OpenAI Aftermath

The Fontaine–Mazur conjecture at the prime 2 concerns two-dimensional 2-adic representations of the absolute Galois group of the rational numbers. In its odd, regular form over Q, the conjecture predicts that every continuous, irreducible, odd representation that is unramified outside finitely many primes and de Rham at p with distinct Hodge–Tate weights comes from a classical cuspidal eigenform, up to a Tate twist. Three manuscripts dated between 23 September and 6 October 2026 in OpenAI's public mathematics collection (github.com/openai/math) claim this statement at p = 2 with no hypothesis on the residual representation. They also claim an occurrence theorem in the completed Hecke algebra without any de Rham hypothesis, and the p = 2 case of Emerton's dimension conjecture for the full Hecke algebra of odd level. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

Background

  • As recounted in the introductions of the manuscripts:
    • Fontaine and Mazur formulated the conjecture in 1995. The deformation-theoretic approach of Wiles and Taylor–Wiles compares Galois deformation rings with Hecke algebras, and the proof of Serre's conjecture by Khare–Wintenberger and Kisin gives modularity of odd, absolutely irreducible residual representations, including in characteristic two.
    • Skinner and Wiles developed ordinary families and pro-modularity arguments for residually reducible representations; Allen adapted the family method to nearly ordinary dyadic representations with solvable residual image; Kisin brought local deformation geometry and the p-adic Langlands correspondence into the problem; Emerton separated occurrence in completed cohomology from the production of classical vectors.
    • At odd primes, Pan established the regular conjecture apart from one local residual configuration at 3, and Xinyao Zhang removed that configuration. At the prime 2, Tung proved a modularity lifting theorem whose global residual image is nonsolvable.
    • Thorne's 2026 preprint proves that ordinary, potentially crystalline representations of Hodge–Tate weights {0, 1} are modular at every prime, without a residual irreducibility hypothesis; the manuscripts use this as an essential input.
    • Emerton formulated the conjecture that every irreducible component of the full p-adic Hecke algebra has Krull dimension four (his Conjecture 2.9); his Corollary 2.28 gives the lower bound of four used by the third manuscript.

The original results

  • Fontaine–Mazur modularity at the prime 2 (23 September 2026; revised version 6 October 2026). Theorem 1.1 of the manuscript states that every continuous, irreducible, odd representation of the absolute Galois group of Q into GL2 over the algebraic closure of Q2, unramified outside finitely many primes and de Rham at 2 with distinct Hodge–Tate weights, is a Tate twist of the 2-adic Galois representation of a classical cuspidal eigenform. The manuscript states that this resolves the odd, regular two-dimensional conjecture over Q at 2, including every semisimple residual representation, scalar and reducible ones among them, and that together with the odd-prime results it gives the modularity conclusion at every prime. It also states that the theorem supplies the weight-two modularity input of the collection's manuscript on Hilbert's tenth problem over the rational numbers.
  • Unrestricted pro-modularity at the prime two (4 October 2026; revised version 6 October 2026). Theorem 1.1 of the second manuscript states that every continuous, odd, absolutely irreducible two-dimensional 2-adic representation of the absolute Galois group of Q, unramified outside finitely many primes, occurs in the full completed Hecke algebra of some odd tame level; the level may contain auxiliary primes. The manuscript states that this needs no de Rham hypothesis and does not assert classical modularity, and it relates the statement to Emerton's expectation for the full varying-weight Hecke algebra, which prescribes the allowed tame primes.
  • The dimension of the two-adic Hecke algebra at odd level (5 October 2026). Theorem 1.1 of the third manuscript states that for every odd positive integer N every irreducible component of the full 2-adic Hecke algebra of level Γ1(N), with all weights, Eisenstein systems and residual systems included, has Krull dimension exactly four. It states that this is the p = 2 case of Emerton's Conjecture 2.9.
    • Method, as described: the first manuscript proves potential occurrence of the target and of auxiliary families in one completed Hecke algebra of definite quaternionic forms over a solvable totally real field split completely at 2, by patching at a characteristic-two curve with no local condition at 2 and by a connectedness argument that records invariant lines at some dyadic places. It then produces a classical vector, through the p-adic local Langlands correspondence when the local representation at 2 is irreducible and through an ordinary family whose dense classical points are made modular by Thorne's theorem when it is reducible, and descends along cyclic steps. The second manuscript finds a regular de Rham point in a family through the given representation, applies the first theorem there, and transfers Hecke support using the dimension theorem and the adjoint Selmer vanishing theorem of Newton and Thorne. The third combines Emerton's lower bound, Chenevier's determinant laws at 2, a bound on the Eisenstein locus and the Newton–Thorne vanishing theorem.
  • Sources. Commit 1581d4c of the openai/math repository, folders Fontaine–Mazur modularity at the prime 2 (6 October 2026) and its earlier version (23 September 2026), Unrestricted pro-modularity at the prime two (6 October 2026) and its earlier version (4 October 2026), and The dimension of the two-adic Hecke algebra at odd level.
  • Formalization. The collection has no Lean scope note in lean/docs/ for this family.

Public follow-up work

Date Work Relation Public claim Public status (checked 17 October 2026)
2026-10-16 J. Jardine, Lattice24, referee reviews rank_249, rank_302 and rank_521 in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) Review (AI-assisted audit) The three reviews, of the 6 October versions of the first two manuscripts and of the third, report no gap. The review of the first notes that it did not fully verify the propagation and connectedness machinery behind one family step, and that it could not check the exact hypotheses and sign convention of Thorne's theorem. The review of the third notes that Emerton's lower bound is used as a black box at 2, apart from the Noetherianity input that the manuscript reproves. The review of the second notes that it depends on the other two manuscripts. Part of a completed audit of the collection's manuscripts without Lean; the reviews are published under CC BY 4.0.

Notes

  • For the prime 3 the first manuscript cites Xinyao Zhang's arXiv:2412.06812v4 of 22 July 2025. A fifth version appeared on 7 October 2026, marked as a major change; its acknowledgments thank GPT 6 Astra "for pointing out a serious error in an earlier version" of one proposition and for help in modifying the strategy of the proof in one section.
  • The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
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