View source for Modularity of elliptic curves over imaginary quadratic fields

'''Modularity of elliptic curves over imaginary quadratic fields''' is a manuscript dated 4 October 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]). This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper or a manuscript is not presented here as independent verification.

== The original result ==
* '''Claim.''' Every elliptic curve over every imaginary quadratic field is modular, with matching local parameters at every place (summary in the collection's contents list).
* '''Source.''' [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Modularity-of-elliptic-curves-over-imaginary-quadratic-fields-October-4-2026 Manuscript folder] in the openai/math repository (commit <code>adc7f12</code>). The collection has no Lean formalization of this result.
* '''Earlier published result.''' A. Caraiani and J. Newton, ''On the modularity of elliptic curves over imaginary quadratic fields'' ([https://arxiv.org/abs/2301.10509 arXiv:2301.10509], v3, 27 March 2025), prove that every elliptic curve is modular over infinitely many imaginary quadratic fields, including '''Q'''(√−''d'') for ''d'' = 1, 2, 3, 5; the main condition is that the modular curve ''X''<sub>0</sub>(15) has rank 0 over the field.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 13 October 2026)
|-
| 2026-10-07; revised 2026-10-09 || CaptainSude, [https://github.com/CaptainSude/elliptic-curves-cm-modularity ''Every elliptic curve over a CM number field is modular''] (releases [https://github.com/CaptainSude/elliptic-curves-cm-modularity/releases/tag/v1.0.0 v1.0.0] and [https://github.com/CaptainSude/elliptic-curves-cm-modularity/releases/tag/v1.0.1 v1.0.1]) || Extension || Every elliptic curve over every CM number field is modular, including CM fields that contain fifth roots of unity and non-Galois CM fields. The abstract states that the proof combines OpenAI's covering construction with a mod-5 automorphy lifting argument that retains a residually ramified Steinberg place, following a method of Taylor and Khare–Thorne. || A 15-page manuscript with LaTeX source. The README reports an AI-assisted internal audit and a team review, and states that this "does not constitute formal verification or independent peer review". Version 1.0.1 revises attribution only; the README states that the theorem statements and proofs are unchanged.
|-
| 2026-10-08 || Bao Le Hung (contributor), [https://hexagonmath.org/2610.00156v1 ''Modularity of elliptic curves over totally real field and CM fields not containing ζ<sub>5</sub>''], Hexagon 2610.00156v1 || Extension; analysis || Notes on the structure of OpenAI's argument. According to the abstract, the argument also covers all totally real fields and, in its original form, CM fields not containing ζ<sub>5</sub>; the totally real case is treated separately to isolate the new idea, which the author describes as orthogonal to modularity lifting theorems. The notes cite the CaptainSude manuscript for the general CM case. || Posted on Hexagon (v1). The page labels the text "a mix of human-written and AI-generated text" with human understanding of "some parts", and states that the first draft was generated by Astra GPT 6.1, with the introduction written by the contributor and revisions by Claude Opus 5.5.
|}

== Notes ==
* Both follow-ups state substantial AI assistance in producing their texts.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]