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Modularity of elliptic curves over imaginary quadratic fields

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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.

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

The paper states

Theorem 1.1

In Modularity of elliptic curves over imaginary quadratic fields, 4 October 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Let K be an imaginary quadratic number field and E/K an elliptic curve. There is an isobaric automorphic representation \pi_E of \GL_2(\A_K) with trivial central character and weight zero such that, for every prime l and every isomorphism \iota:\overline\Q_l\xrightarrow{\sim}\C,

r_\iota(\pi_E)\simeq H^1_{\et}(E_{\overline K},\overline\Q_l).

At every finite place v\nmid l one has

\iota\WD\bigl(H^1_{\et}(E_{\overline K},\overline\Q_l) |_{G_{K_v}}\bigr)^{\Fss} \simeq \rec_{K_v}\bigl(\pi_{E,v}\otimes|\det|_v^{-1/2}\bigr).

The unitary parameter at the complex place is z\mapsto\diag(z/|z|,\overline z/|z|). In particular every local factor agrees, and

L(E/K,s)=L(\pi_E,s-1/2).

If E has no complex multiplication, \pi_E is cuspidal. In the CM case it is given by automorphic induction of a Hecke character, or by an isobaric sum when the CM field is already K.

Reported proof verification
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Papers and explanations
One current paper, readable here as a page beside its PDF. Two explanations by others.

Papers

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Lean statements

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Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

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Explanations by others

Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.

  • Hexagon, Hexagon: a repository for research in the mathematical sciences and theoretical computer science, 8 October 2026. Source

    hexagon:2610.00156v1 (Bao Le Hung) parses the structure of OpenAI’s modularity argument for elliptic curves over imaginary quadratic fields, also covering totally real fields and CM fields without a fifth root of unity.

  • Frank Calegari, A Brief Digest of Some New OpenAI Results, 8 October 2026. Source

    Caraiani and Newton had nearly proved modularity over imaginary quadratic fields; OpenAI overcame the reducible mod 3 and mod 5 case with a trick, and Bao Le Hung’s Hexagon preprint shows it also gives totally real fields and CM fields without zeta_5.

What these papers cite in the release, and what cites them

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Papers elsewhere in the release that this family’s papers cite

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Papers elsewhere in the release that cite this family’s papers

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References outside the release

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What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.

    Author not stated. Source

  • Kevin Buzzard, Algebraic number theory results without Lean, 7 October 2026. Source

    An expert flags that the release gives no Lean for modularity of elliptic curves over imaginary quadratic fields and suggests it may be the last mile of a programme already close to done rather than a new idea.

  • Frank Calegari, The OpenAI problem dump, 6 October 2026. Source

    Update says modularity of elliptic curves over imaginary quadratic fields is in the dump; a commenter reports the method also gives totally real and apparently CM fields.

Questions awaiting review

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Identify the theorem the catalogue describes

  • Family 030 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

State the result in Lean

  • Family 030 The release has no Lean statement for this family. Open the form