Serre modularity over real quadratic fields, primarily AI-generated
Awaiting review
The takeaway
Dieulefait posted two Claude-assisted Fontaine–Mazur modularity papers the same day.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Unverified claim
- Importance
- 3 of 5
- Last verified
- 5 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 2611.00071: d ∈ {2,3,5,6,7,10,11,13,17,21,33,37,41,57,65,73,89}; every continuous, absolutely irreducible, totally odd 2-dim mod-p (p odd) representation is modular; d = 17 unconditional
- Provenance: ‘Primarily AI-generated’; human understanding ‘some’; ‘the authors used ChatGPT 6-Astra and Sol 6.1 as an assistant in mathematical research and manuscript preparation … exploring proof strategies, checking mathematical arguments …’
- Hexagon 2611.00067 (Dieulefait): Fontaine–Mazur for residually reducible 2-adic representations over totally real fields where 2 splits, generalizing an OpenAI result; ‘produced with the assistace of Claude’
- Hexagon 2611.00096 (Dieulefait): Fontaine–Mazur in the residually reducible mixed case via fully ordinary families; ‘the outcome of an interaction with Claude’
What happened
Three Hexagon deposits of 4 Nov 2026 come from number theorists who work on modularity. The headline claim, 2611.000711, extends Serre’s conjecture, proved over Q by Khare–Wintenberger, to 17 real quadratic fields, conditionally on GRH. Its new ingredients are a modularity criterion for 2-adic representations with scalar reduction and integral trace arguments. The authors classify all three works as primarily AI-generated and only partly human-checked.
Why it matters
Serre’s conjecture beyond Q is open in general. The conditional result covers specific fields, so it is not the general conjecture. Hexagon does not judge correctness, and no expert reaction had been found as of 5 Nov.
What is disputed or not yet verified
| Verification | Hexagon deposits, unrefereed |
|---|
Sources
3 sources from 1 site. Numbers match the chips in the text.
3 sources: 3 primary
Primary
- Hexagon 2611.00071: Serre modularity over real quadratic fieldshexagonmath.org, paper
- Hexagon 2611.00067: Fontaine–Mazur for 2-adic representationshexagonmath.org, paper
- Hexagon 2611.00096: Fontaine–Mazur, residually reducible mixed casehexagonmath.org, paper
Changes
- Filed from the Hexagon records