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Math-75-A2 · revision r1 · on Math-75

OpenAI claim summary for Math-75

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 modularity construction applies to imaginary quadratic base fields. This record ranges over every number field under its stated endomorphism condition, leaving the other base fields untreated.

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

Manuscript

Contents
  1. Modularity of Elliptic Curves over Number Fields (End_K(E)=Z)
  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

Modularity of Elliptic Curves over Number Fields (End_K(E)=Z)

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 030; 2026-10-04), the asserted result is that all elliptic curves over imaginary quadratic fields are modular with the stated local-parameter compatibility at every place.

Mathematical statement of the claimed result

Manuscript 1.
For every imaginary quadratic number field $K$ and elliptic curve $E/K$, the manuscript claims an isobaric automorphic $\pi_E$ on $\operatorname{GL}_2(\mathbb A_K)$ of trivial central character and weight zero with, for every prime $\ell$ and $\iota:\overline{\mathbb Q}_{\ell}\simeq\mathbb C$, \[ r_{\iota}(\pi_E)\simeq H^1_{\rm et}(E_{\overline K},\overline{\mathbb Q}_{\ell}),\qquad L(E/K,s)=L(\pi_E,s-1/2). \] For every finite $v\nmid\ell$, the Frobenius-semisimplified Weil--Deligne representation is claimed to equal local Langlands for $\pi_{E,v}\otimes|\det|_v^{-1/2}$. The complex-place unitary parameter is $\operatorname{diag}(z/|z|,\bar z/|z|)$. When $E$ has no CM the asserted representation is cuspidal. For CM curves it is induced automorphically, with a sum of characters instead when $K$ already contains the CM field. All local factors, including bad-reduction factors, are included.

Outline of the argument

For manuscript 1 (family 030; 2026-10-04), cyclic covers have one degeneration whose Jacobian splits into copies of the target curve and another that makes monodromy explicit. A geometric switch of coefficient primes and three automorphic lifts propagate automorphy through the family before descent to the original field.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 030; 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