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Modularity of elliptic curves over imaginary quadratic fields

Let KK be an imaginary quadratic field and E/KE/K an elliptic curve. The Langlands program predicts that EE is modular: there is an automorphic representation πE\pi_E of GL2(AK)\mathrm{GL}_2(\mathbb A_K) (a Bianchi modular form when EE has no complex multiplication) whose Galois representations are Het1(EKˉ,Qˉl)H^1_{et}(E_{\bar K},\bar{\mathbb Q}_l), so that L(E/K,s)=L(πE,s−1/2)L(E/K,s)=L(\pi_E,s-1/2). Over Q\mathbb Q this is the theorem of Wiles, Taylor-Wiles and Breuil-Conrad-Diamond-Taylor. Over imaginary quadratic fields, potential modularity was known (Allen et al., 2023), Allen-Khare-Thorne proved modularity for a positive proportion of curves, and Caraiani-Newton for all curves over fields where X0(15)(K)X_0(15)(K) has rank zero. Is every elliptic curve over every imaginary quadratic field modular?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Langlands program: modularity of elliptic curves
Posed by
Robert P. Langlands, Problems in the theory of automorphic forms, LNM 170 (1970), Section 7, as the manuscript cites; studied computationally by Grunewald-Helling-Mennicke and Cremona
Year posed
1970
Years open
56y
Solved
2026-10-04
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
58 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every imaginary quadratic KK and every E/KE/K there is an isobaric automorphic representation πE\pi_E of GL2(AK)\mathrm{GL}_2(\mathbb A_K) with trivial central character and weight zero such that rι(πE)≃Het1(EKˉ,Qˉl)r_\iota(\pi_E)\simeq H^1_{et}(E_{\bar K},\bar{\mathbb Q}_l) for every ll and ι\iota, with local-global compatibility at every finite place v∤lv\nmid l and the expected parameter at the complex place; hence L(E/K,s)=L(πE,s−1/2)L(E/K,s)=L(\pi_E,s-1/2). πE\pi_E is cuspidal when EE has no CM. The method is a geometric prime switch: cyclic covers of a quadratic twist of EE give a chain of mod pp, mod 3, mod pp congruences that carries automorphy from an auxiliary curve to EE through the Caraiani-Newton lifting theorem. It does not treat other CM fields or totally real fields.

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The README also cautions that unformalized results could have issues.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, read against the modularity conjecture for imaginary quadratic fields as the manuscript frames it; the proof was not refereed. The argument uses deep published and preprint inputs as black boxes, above all the Caraiani-Newton lifting theorem and residual-image criterion (arXiv:2301.10509v3, cited as a 2025 preprint) and the potential automorphy work of Allen et al.; the manuscript says Caraiani-Newton remains essential, so the result stands on those inputs as specialized here. The release has no Lean formalization for this family, and the README cautions that unformalized results could have issues.

Sources

Changelog1 change

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