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Math-2807 · mathematics · Number theory

Unrestricted odd two-adic pro-modularity over Q

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every continuous absolutely irreducible odd representation r of Gal(Qbar/Q) in GL2(E), where E is a finite extension of Q2, unramified outside finitely many finite primes, is there an odd level N containing every odd ramified prime and a continuous Z2-algebra map from the full prime-to-2N, all-weight level-Gamma1(N) Hecke algebra to the integers of E that sends $T_l$ to $\operatorname{tr}(r(Frob_l))$ and l $S_l$ to $det(r(Frob_l))$ at every l not dividing 2N? No de Rham condition is imposed.

Short English statement

For every continuous absolutely irreducible odd representation r of Gal(Qbar/Q) in GL2(E), where E is a finite extension of Q2, unramified outside finitely many finite primes, is there an odd level N containing every odd ramified prime and a continuous Z2-algebra map from the full prime-to-2N, all-weight level-Gamma1(N) Hecke algebra to the integers of E that sends $T_l$ to $\operatorname{tr}(r(Frob_l))$ and l $S_l$ to $det(r(Frob_l))$ at every l not dividing 2N? No de Rham condition is imposed.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2807-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2807

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; 2026-10-04).

Unrestricted odd two-adic pro-modularity over Q

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; 2026-10-04), the asserted result is that every odd, absolutely irreducible two-dimensional two-adic representation of the rational Galois group with finite ramification occurs in completed Hecke theory at some odd tame level, without a de Rham assumption.

Mathematical statement of the claimed result

Manuscript 1.
Let $E/\mathbb Q_2$ be finite and let $r:G_{\mathbb Q}\to\operatorname{GL}_2(E)$ be continuous, absolutely irreducible, unramified outside finitely many primes, and odd, meaning $\det r(c)=-1$ for complex conjugation. No de Rham or Hodge--Tate condition is imposed. The claim provides an odd integer $N$, divisible by every odd ramified prime, and a continuous $\mathbb Z_2$-algebra homomorphism \[ \lambda:\mathbb T_2(N)\longrightarrow\mathcal O_E, \qquad\lambda(T_{\ell})=\operatorname{tr}r(\operatorname{Frob}_{\ell}),\quad \lambda(\ell S_{\ell})=\det r(\operatorname{Frob}_{\ell})\quad(\ell\nmid2N). \] The full algebra includes Eisenstein forms. Its definition is \[ \mathbb T_2(N)=\varprojlim_k\left(\mathbb Z_2\otimes_{\mathbb Z}T_{\le k}^{(2)}(N)\right), \] where $T_{\le k}^{(2)}(N)$ is the $\mathbb Z$-algebra of operators on $\bigoplus_{i=1}^kM_i(\Gamma_1(N))$ generated by $T_\ell$ and $\ell S_\ell$ for primes $\ell\nmid2N$. Here $M_i$ denotes all holomorphic modular forms of weight $i$, and on that weight $S_\ell=\ell^{i-2}\langle\ell\rangle$. Each stage has the two-adic topology and the limit has its inverse-limit topology. The conclusion is pro-modularity, expressed by this compatible completed-Hecke eigensystem.

Outline of the argument

For manuscript 1 (family 010; 2026-10-04), a trace family is supplied with potentially automorphic support and a regular point. Coinvariants of local block modules are compared along characteristic-two curves; admissibility and a full adjoint tangent estimate propagate the supporting component back to the original representation.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 010; 2026-10-04).

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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