OpenAI: elliptic curves over imaginary quadratic fields are modular
Awaiting review
Importance: major (4 of 5)The takeaway
Family 030 of OpenAI’s 6 Oct 2026 release is a 30-page manuscript proving that every elliptic curve over every imaginary quadratic field is modular. Caraiani and Newton had done this for a positive proportion of fields; the new step handles curves whose mod-3 and mod-5 representations are both reducible. Frank Calegari calls it ‘a really nice trick rather than a fundamental breakthrough’. Within two days Bao Le Hung posted a note saying the same idea also proves modularity over all totally real fields, itself a long-open problem. Nothing is formalized or refereed.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- Major (4 of 5)
- Last verified
- 29 October 2026
Your AI and this story
- GPT-6 Astra159 days after its cutoff
- Claude Opus 5.598 days after its cutoff
- Gemini 4 Argon189 days after its cutoff
- Grok 4.7128 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 98 days before it.
Key facts
- Abstract (verbatim): ‘We prove the modularity conjecture for elliptic curves over imaginary quadratic fields: every such curve is modular, with matching local parameters at every place.’ 30 pages, dated 4 Oct 2026. Theorem 1.1 gives an isobaric automorphic representation of GL2(A_K), cuspidal when E has no CM, with L(E/K, s) = L(π_E, s − 1/2)
- Prior state (paper, §1.1): Allen–Khare–Thorne proved modularity for a positive proportion of curves over CM fields not containing ζ5; Caraiani–Newton proved ‘modularity of every elliptic curve over an imaginary quadratic field K for which X0(15)(K) has rank zero’. Verbatim: ‘Theorem 1.1 removes the remaining restrictions on the curve and the imaginary quadratic field. Caraiani–Newton’s lifting theorem and residual-image criterion for modularity of elliptic curves remain essential inputs.’
- New ingredient (paper, §1.2): a ‘geometric prime switch’ that constructs the intermediate representations ‘as character pieces of a single family of Jacobians’, so that a known modularity theorem applies while control is kept at the primes used for lifting
- Frank Calegari (Persiflage, 8 Oct, verbatim): ‘This problem was almost already solved by Caraiani and Newton … The problem is what to do with elliptic curves which, for example, have reducible mod 3 and mod 5 representations. This is the problem that OpenAI overcame, by what I think is a very nice trick.’ And: ‘I view this result as definitely something a human could come up with. It’s a really nice trick rather than a fundamental breakthrough.’
- On the release night Calegari scored his own list of ten open problems: ‘As BLH says, modularity of elliptic curves over imaginary quadratic fields is in, for 2 points’ (update to ‘The OpenAI problem dump’, 6 Oct)
- Bao Le Hung (Hexagon 2610.00156, 8 Oct, verbatim): ‘The argument also proves modularity over all totally real field, and in the original form, modularity over CM fields not containing ζ5.’ He keeps the totally real case separate ‘to highlight the main new idea in the OpenAI proof, which is orthogonal to modularity lifting theorems’. Provenance: ‘First draft generated by Astra GPT 6.1, introduction written by the contributor, revisions by Claude Opus 5.5’. Before this, modularity over totally real fields was known for real quadratic, cubic and some quartic fields
- A GitHub manuscript by ‘CaptainSude’ (7 Oct, v1.0.1 on 9 Oct) claims modularity over every CM field, including those containing ζ5, taking the OpenAI paper as a principal input; Toby Gee commented that its new idea at p = 5 ‘is already in the literature, due to Khare--Thorne … and Taylor’, and the attribution was revised. Elliot Glazer said its similarity to the improvement discussed on Calegari’s blog increased his credence that the OpenAI result is correct
What happened
An elliptic curve is modular when its L-function comes from an automorphic form; over ℚ this is the theorem of Wiles, Taylor–Wiles and Breuil–Conrad–Diamond–Taylor behind Fermat’s Last Theorem. Over imaginary quadratic fields such as ℚ(i) the corresponding automorphic forms live on hyperbolic three-space, and only recently (Allen–Khare–Thorne, Caraiani–Newton) did modularity become provable for many curves. Caraiani and Newton covered all curves over a field K whenever the modular curve X0(15) has rank zero over K, which happens for a positive proportion of fields. Curves whose mod-3 and mod-5 representations are both reducible were the main remaining obstacle, as Calegari describes it.
OpenAI’s 30-page manuscript proves modularity for every curve over every imaginary quadratic field. It keeps Caraiani–Newton’s lifting theorem and adds a new “prime switch”: instead of Wiles’s single auxiliary curve, it builds the intermediate Galois representations from character pieces of one family of Jacobians.
Why it matters
Modularity over number fields beyond ℚ is a central goal of the Langlands program and an input to many Diophantine applications. If the manuscript is right, a two-day-old extension already reaches the totally real case. Nobody has yet published a complete check, and there is no formal proof.
What is disputed or not yet verified
| Verification | Unrefereed manuscript, no Lean formalization; expert readings on Persiflage judge the key trick sound but do not claim a full check |
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Reactions
Number theorists engaged quickly on Frank Calegari’s blog. Calegari called the step “a very nice trick” that a human could have found, not “a fundamental breakthrough”. Bao Le Hung, a specialist in modularity lifting, posted a note (first draft by GPT-6.1 Astra, edited by him and Claude) saying that the same idea proves modularity over every totally real field, a problem settled before only for real quadratic, cubic and some quartic fields, and over CM fields not containing fifth roots of unity. A further GitHub manuscript claims all CM fields; Toby Gee pointed out that its key idea at the prime 5 was already in the literature.
Sources
8 sources from 4 sites. Numbers match the chips in the text.
8 sources: 3 primary, 5 reactions
Primary
- OpenAI: Modularity of elliptic curves over imaginary quadratic fields (4 Oct)github.com, paper
- Hexagon 2610.00156: Bao Le Hung, Modularity of elliptic curves over totally real fields and CM fields not containing ζ5hexagonmath.org, paper
- GitHub CaptainSude/elliptic-curves-cm-modularity: Every Elliptic Curve over a CM Number Field Is Modulargithub.com, code
Reactions
- Persiflage (Frank Calegari): A Brief Digest of Some New OpenAI Results (8 Oct)galoisrepresentations.org, discussion
- Persiflage: The OpenAI problem dump (6 Oct, with update and comments)galoisrepresentations.org, discussion
- Toby Gee’s GitHub issue on the CM follow-up (citations to Khare–Thorne and Taylor)github.com, discussion
- Elliot Glazer on X (8 Oct)x.com, discussion
- oai-math-revised: community notes on the modularity manuscriptgithub.com, discussion
Changes
- Filed (families 001–125; manuscript introduction, Persiflage posts and comments, Le Hung’s Hexagon page, the CaptainSude repository issue and Glazer’s post read)