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Math-2807-A1 · revision r1 · on Math-2807

OpenAI claim summary for Math-2807

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

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).

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Unrestricted odd two-adic pro-modularity over Q
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

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