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Modularity of elliptic curves over imaginary quadratic fields

OpenAI · October 4, 2026

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Abstract

We prove the modularity conjecture for elliptic curves over imaginary quadratic fields: every such curve is modular, with matching local parameters at every place.

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Notes

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work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-23) and not checked by a person. May be wrong or incomplete. Report an error.

Follow-up works

Comments and discussion

  • Persiflage and commenters, The OpenAI problem dump (galoisrepresentations.org, 6 Oct 2026). In the comments, BLH says the paper's "new chaining argument" seemed strong enough to give the general totally real case and the CM case not containing ζ₅; Persiflage's update counts the result as "in, for 2 points".
  • Persiflage, A Brief Digest of Some New OpenAI Results (galoisrepresentations.org, 8 Oct 2026). Item 1 places the paper relative to Caraiani–Newton, identifies the treatment of curves with reducible mod 3 and mod 5 representations as the new step, and discusses Le Hung's proposed extensions; it calls the method "a really nice trick rather than a fundamental breakthrough".
  • toby-gee, Citations (GitHub issue #1, 9 Oct 2026). Discussing the CM-field follow-up, the commenter traces its characteristic-five exceptional-cohomology mechanism to Taylor and Khare–Thorne, prompting an attribution revision.
  • Elliot Glazer (@ElliotGlazer), post on X (8 Oct 2026). Says a claimed extension of the result resembles "a weaker improvement of OAI's result" discussed in BLH's comment thread on Frank Calegari's blog, which increases his credence that it is correct.
  • Thelonious Oditamo (@originaldint), post on X (8 Oct 2026). A two-post thread arguing that the proof "works verbatim for any field not containing zeta_5" and that the paper understates its contribution, and contrasting this with a manuscript claiming a mod 5 modularity lifting result for the remaining CM fields.

Formalization

Possibly relevant literature (found by the agent, not checked)

Elsewhere

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Cite

@misc{OAI:Modularity-of-elliptic-curves-over-imaginary-quadratic-fields-October-4-2026,
  author = {{OpenAI}},
  title = {{Modularity of elliptic curves over imaginary quadratic fields}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Modularity-of-elliptic-curves-over-imaginary-quadratic-fields-October-4-2026/paper.pdf}{OAI:Modularity-of-elliptic-curves-over-imaginary-quadratic-fields-October-4-2026}},
  year = {2026}
}

Family 030: Modularity of elliptic curves over imaginary quadratic fields

Proves the modularity conjecture for elliptic curves over imaginary quadratic fields: every elliptic curve over every imaginary quadratic field is modular, with matching local parameters at every place.

This is the only paper in this family.