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.

Manuscripts (3)

Unrestricted pro-modularity at the prime two

October 6, 2026 · source · cite as \cite{OAI:Unrestricted-pro-modularity-at-the-prime-two-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.

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

October 5, 2026 · source · cite as \cite{OAI:The-Dimension-of-the-Two-Adic-Hecke-Algebra-at-Odd-Level-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.

Fontaine–Mazur modularity at the prime 2

October 6, 2026 · source · cite as \cite{OAI:Fontaine-Mazur-modularity-at-the-prime-2-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.

Result 010 · Number theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.