---
id: "2026-11-04-serre-modularity-real-quadratic-fields-chatgpt-astra"
url: "https://postcutoff.com/e/2026-11-04-serre-modularity-real-quadratic-fields-chatgpt-astra/"
as_of: "2026-11-09T21:13:00+01:00"
date: "2026-11-04"
date_precision: day
category: science
importance: 3
confidence: medium
status: [Awaiting review]
verification: Unverified claim
sources: 3
editor: human editor
human_review: null
version: "2026-11-05"
---

As of: 2026-11-09 21:13 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-11-04-serre-modularity-real-quadratic-fields-chatgpt-astra/

# Serre modularity over real quadratic fields, primarily AI-generated

Full title: Dieulefait, Freitas and Pacetti post a primarily AI-generated proof of Serre's modularity conjecture over 17 real quadratic fields (under GRH)

On Hexagon, Luis Dieulefait, Nuno Freitas and Ariel Pacetti claim Serre's modularity conjecture over Q(√d) for 17 values of d assuming GRH for Dedekind zeta functions (d = 17 unconditional), in a work declared "primarily AI-generated" with ChatGPT 6-Astra and Sol 6.1. Dieulefait posted two Claude-assisted Fontaine–Mazur modularity papers the same day. All are unrefereed.

## 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, [2611.00071](https://hexagonmath.org/abs/2611.00071), 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

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 188 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 127 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 218 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 157 days after its cutoff

## Sources

1. [Hexagon 2611.00071: Serre modularity over real quadratic fields](https://hexagonmath.org/abs/2611.00071) (hexagonmath.org, paper)
2. [Hexagon 2611.00067: Fontaine–Mazur for 2-adic representations](https://hexagonmath.org/abs/2611.00067) (hexagonmath.org, paper)
3. [Hexagon 2611.00096: Fontaine–Mazur, residually reducible mixed case](https://hexagonmath.org/abs/2611.00096) (hexagonmath.org, paper)

## Changes

- 2026-11-05 (filed): Created from the Hexagon records

## Related

- 2026-10-28: [Kisin proves Serre's 1960 Artin-character conjecture with GPT Astra](https://postcutoff.com/e/2026-10-28-kisin-serre-artin-character-conjecture-gpt-astra/index.md)