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.
released 2026-10-06 | 2 theorems · 10 lemmas · 19 proofs · 14,923 words |
PLAY LEVEL 1 »(pdf)
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.
released 2026-10-05 | 2 theorems · 4 lemmas · 10 proofs · 5,114 words |
PLAY LEVEL 2 »(pdf)
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.
released 2026-10-06 | 3 theorems · 29 lemmas · 41 proofs · 36,899 words |
PLAY LEVEL 3 »(pdf)
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.