Elliptic-curve torsion in degrees 6–9, mostly by GPT-6.1 Sol
Awaiting review
Importance: major (4 of 5)The takeaway
Filip Najman deposited on Hexagon a paper that determines every group occurring as the torsion of an elliptic curve over a number field of degree 6, 7, 8 or 9, including three groups not previously known to occur.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Unverified claim
- Importance
- Major (4 of 5)
- Last verified
- 6 November 2026
Your AI and this story
- GPT-6 Astra188 days after its cutoff
- Claude Opus 5.5127 days after its cutoff
- Gemini 4 Argon218 days after its cutoff
- Grok 4.7157 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 127 days before it.
Key facts
- Hexagon 2611.00073, 4 Nov 2026; authorship mode ‘no-human-author-claimed’, human understanding ‘some’; tools ChatGPT 6.1 Sol, Claude Opus 5.5, Magma
- Sporadic groups beyond the infinite families: 4 in degree 6, 3 in degree 7, 8 in degree 8, 9 in degree 9; Z/2×Z/24 (sextic), Z/3×Z/15 (octic) and Z/2×Z/26 (nonic) were not known to occur
- For 13 of the 24 (degree, group) pairs the paper also determines all elliptic curves realizing them
- Methods: Derickx’s global method, Hecke sieves, formal immersion at 2, Riemann–Roch on modular curves over finite fields
- Provenance: ‘OpenAI’s GPT-6.1 Sol produced almost all of the mathematics of the first versions of the proofs … starting in mid-October 2026. Anthropic’s Claude Opus 5.5 reviewed these versions, independently recomputed some of the computations … and wrote the manuscript’
- Najman: ‘I have carefully checked, understood and rewritten the degree 6 part; that version will appear separately on arXiv. Mathematically, the present paper should be regarded as raw data’
What happened
Filip Najman, a specialist in elliptic-curve torsion, deposited Hexagon 2611.000732 on 4 Nov 2026. Its provenance record says GPT-6.1 Sol produced the first proofs for each degree separately, and Claude Opus 5.5 merged and wrote them up. Independent AI agents then reviewed the merged paper. Najman posed the questions, guided the work and made the editorial decisions.
Why it matters
Classifying torsion over fields of fixed degree is a long-running programme (Mazur for d = 1, Kamienny–Kenku–Momose for d = 2, then degree 3 and 4 by Derickx, Etropolski, van Hoeij, Morrow, Najman, Zureick-Brown and others). This deposit claims four further degrees at once. By the author’s own account it is not yet fully human-checked. Hexagon does not judge correctness.
Follow-up (5 Nov)
A day later Najman deposited Hexagon 2611.001201, which determines the set S(d) of primes p occurring as the order of a point on an elliptic curve over a number field of degree d, for every 9 ≤ d ≤ 22 (S(d) was known for d ≤ 8; e.g. the largest prime in S(22) is 67). Provenance: “GPT-6.1 Sol and GPT-6 Astra produced almost all of the mathematics of the first versions of the proofs and carried out the computations, between 28 October and 3 November 2026. Anthropic’s Claude Opus 5.5 reviewed these proofs, without rerunning the computations”. Human understanding: “some”.
What is disputed or not yet verified
| Verification | Hexagon deposit, unrefereed; degree 6 human-checked, degrees 7–9 not yet |
|---|
Sources
2 sources from 1 site. Numbers match the chips in the text.
2 sources: 2 primary
Primary
Changes
- Added the degree 9–22 follow-up (Hexagon 2611.00120)
- Filed from the Hexagon record