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The Dimension of the Two-Adic Hecke Algebra at Odd Level
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Abstract
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.
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Possibly relevant literature (found by the agent, not checked)
- Xinyao Zhang, Modular points and dimensions of Eisenstein deformation spaces (arXiv:2512.21249, v1 Dec 2025, v3 7 October 2026). Theorem 6.4.1 (see Remark 1.2.12) establishes Emerton's equidimensionality conjecture (Conjecture 2.9) for every odd prime, the odd-prime counterpart of this paper's dimension-four claim at $p = 2$.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 010
Independent sites; their content is not reviewed here.
Cite
@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}
}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:
- Unrestricted pro-modularity at the prime two · October 6, 2026
- Fontaine–Mazur modularity at the prime 2 · October 6, 2026