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Claimed AI-led

Modularity of every elliptic curve over every imaginary quadratic field

An OpenAI manuscript claims that every elliptic curve over every imaginary quadratic field is modular, with the expected local parameters at every place, removing the restrictions on the curve and the field left by Caraiani and Newton's theorem.

Model
Unreleased internal OpenAI model, not named in the release
Field
Mathematics
Date
2026-10-06

Sources

Original work

GitHubModularity of elliptic curves over imaginary quadratic fields (PDF)β†—

Announcement

OpenAISharing AI progress in mathematics↗

What was found

The theorem attaches to each curve EE over an imaginary quadratic field KK an isobaric automorphic representation of GL2(𝔸K)\mathrm{GL}_2(\mathbb{A}_K) realizing its cohomology, with local-global compatibility at every finite place and L(E/K,s)=L(Ο€E,sβˆ’1/2)L(E/K, s) = L(\pi_E, s - 1/2); it is cuspidal when EE has no complex multiplication. Over β„š\mathbb{Q} this is the theorem of Wiles, Taylor and Wiles and of Breuil, Conrad, Diamond and Taylor. Over imaginary quadratic fields, potential modularity follows from the ten-author theorem of Allen, Calegari, Caraiani, Gee, Helm, Le Hung, Newton, Scholze, Taylor and Thorne; Allen, Khare and Thorne proved modularity for a positive proportion of curves; and Caraiani and Newton proved it for every curve over fields where X0(15)(K)X_0(15)(K) has rank zero. The manuscript keeps Caraiani and Newton's lifting theorem and residual criterion as inputs and supplies a geometric prime switch, in the spirit of Wiles's 3-5 switch, to reach the curves they could not.

Novelty check

Read the introduction at commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb on 2026-10-12; it cites the line of work above and states that Caraiani and Newton's lifting theorem remains essential. No earlier claim of the full statement was found in the registry. This is an area of intense current human work, so the main novelty risk is not an old result but concurrent unpublished progress by the people working on it, which a registry search cannot see.

Caveats and known objections

No Lean formalization accompanies the family, and an argument of this kind, resting on a deep stack of published automorphy lifting results, is checkable only by the small community that works in it. Nobody outside OpenAI is on record as having read it. Autonomy is graded ai-led for the same reasons as the other families split from this release: humans chose the problems and filtered the output, and the single-agent account rests on OpenAI alone.

Also recorded at

vibemathedmodularity-elliptic-curves-imaginary-quadratic

Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.

Nobody outside the lab has checked this yet.

Reading the primary source closely enough to say whether it supports the claim counts as a check, and you are credited on the entry.

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History of this entry

  1. CorrectedRecorded the vibemathed record for this result as a registration. It cites the same manuscript. No grade moved.
  2. AddedEntered the registry graded Claimed and AI-led.

Entries are never deleted. A grade that does not hold up is downgraded on the record, with the reason beside it.

Community discussion

Hacker NewsSharing AI progress in mathematics↗

Graded claimed for verification and ai-led for autonomy. What these mean.

Cite this entry

Plain text
whataifound.org. (2026). Modularity of every elliptic curve over every imaginary quadratic field. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-10-06-modularity-imaginary-quadratic
BibTeX 
@misc{whataifound-openai-2026-quadratic,
  title        = {Modularity of every elliptic curve over every imaginary quadratic field},
  author       = {{whataifound.org}},
  year         = {2026},
  howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
  note         = {Result by OpenAI. Verification: Claimed. Autonomy: AI-led.},
  url          = {https://whataifound.org/finding/2026-10-06-modularity-imaginary-quadratic}
}

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