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Math-2804 · mathematics · Number theory

Dimension of the full two-adic Hecke algebra at odd level

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Fix any odd positive N. For each k take the Z-algebra generated by prime-to-2N Hecke operators $T_l$ and l $S_l$ on the sum of all weight-i level-Gamma1(N) modular forms, including Eisenstein forms, for $1<=i<=k$; here $S_l$ acts in weight i by $l^{i-2}$ times the diamond operator. Complete each algebra by tensoring with Z2 and take their inverse limit under restriction. Must every irreducible component of its spectrum have Krull dimension exactly four?

Short English statement

Fix any odd positive N. For each k take the Z-algebra generated by prime-to-2N Hecke operators $T_l$ and l $S_l$ on the sum of all weight-i level-Gamma1(N) modular forms, including Eisenstein forms, for $1<=i<=k$; here $S_l$ acts in weight i by $l^{i-2}$ times the diamond operator. Complete each algebra by tensoring with Z2 and take their inverse limit under restriction. Must every irreducible component of its spectrum have Krull dimension exactly four?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2804-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2804

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 010; 2026-10-05).

Dimension of the full two-adic Hecke algebra at odd level

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 010; 2026-10-05), the asserted result is that at any odd $\Gamma_1$ level, all irreducible components of the full two-adic Hecke algebra have Krull dimension four.

Mathematical statement of the claimed result

Manuscript 1.
Fix any odd integer $N\ge1$. Let $M_i(N)$ be all complex modular forms of weight $i$ and level $\Gamma_1(N)$, including Eisenstein series. On their direct sum for $1\le i\le k$, let $T_{\le k}^{(2)}(N)$ be the $\mathbb Z$-algebra generated by $T_{\ell}$ and $\ell S_{\ell}$ at primes $\ell\nmid2N$, where $S_{\ell}$ acts as $\ell^{i-2}\langle\ell\rangle$. Form \[ T_2(N)=\varprojlim_k(\mathbb Z_2\otimes_{\mathbb Z}T_{\le k}^{(2)}(N)). \] The claimed conclusion is that every irreducible component of $\operatorname{Spec}T_2(N)$ has Krull dimension exactly four. All residual components are included; there is no cuspidality, slope or residual-irreducibility restriction.

Outline of the argument

For manuscript 1 (family 010; 2026-10-05), integral determinant laws and finite classical stages establish a Noetherian algebra. Cuspidal points on each component give lower bounds, while adjoint Galois cohomology and local quotients bound the tangent dimension from above.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 010; 2026-10-05).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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