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Math-2780-A1 · revision r1 · on Math-2780

OpenAI claim summary for Math-2780

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; 2026-09-23).

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

Manuscript

Contents
  1. Odd regular Fontaine-Mazur over Q at two
  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

Odd regular Fontaine-Mazur over Q at two

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; 2026-09-23), the asserted result is that an odd irreducible two-dimensional two-adic representation over the rationals, finitely ramified and de Rham with unequal Hodge--Tate weights, comes from a classical modular form after a Tate twist.

Mathematical statement of the claimed result

Manuscript 1.
Let $r:G_{\mathbb Q}\to\operatorname{GL}_2(\overline{\mathbb Q}_2)$ be continuous and irreducible. Assume it is odd, unramified outside a finite prime set, and has a de Rham restriction at $2$ whose two Hodge--Tate weights differ. The claimed modularity conclusion is \[ r\simeq r_{f,2}(m) \] for some classical cuspidal eigenform $f$ and integer Tate twist $m$. No residual irreducibility or prescribed local ordinary condition is part of the displayed theorem.

Outline of the argument

For manuscript 1 (family 010; 2026-09-23), local deformation charts and solvable base change create a family with modular support. Integral completed forms and their block action permit patching, after which an ordinary vector and the prescribed local line give classicality and descent.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 010; 2026-09-23).

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