--- id: "2026-11-09-non-split-cartan-121-two-ai-proofs" url: "https://postcutoff.com/e/2026-11-09-non-split-cartan-121-two-ai-proofs/" as_of: "2026-11-13T13:49:00+01:00" date: "2026-11-09" date_precision: day category: science importance: 4 confidence: medium status: [Awaiting review] verification: Unverified claim sources: 4 editor: human editor human_review: null version: "2026-11-10" --- As of: 2026-11-13 13:49 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/2026-11-09-non-split-cartan-121-two-ai-proofs/ # X_ns+(121): two AI-generated proofs in two days Full title: Two independent, mostly AI-generated proofs that the non-split Cartan modular curve X_ns+(121) has only CM rational points, completing the 11-adic images of Galois On 9 and 10 Nov 2026 two Hexagon deposits claimed, by different methods, that the genus-511 modular curve X_ns+(121) has exactly seven rational points, all CM. With Rouse, Sutherland and Zureick-Brown's earlier work this would finish the classification of 11-adic Galois images of non-CM elliptic curves over Q. Both papers state the mathematics was produced mainly by AI models and neither has been fully checked by a human; one is co-authored by Zureick-Brown, whose conjecture it settles. ## Key facts - Hexagon 2611.00231 (9 Nov): Santiago Arango-Piñeros (UMass Amherst), Filip Najman (Zagreb), David Zureick-Brown (Amherst College); descent via X_ns+(121) → X_ns+(11) (rank-1 elliptic curve), Mordell–Weil sieve, 7 genus-5 hyperelliptic curves over a degree-15 field, Stoll's Selmer-group Chabauty - Provenance (2611.00231): 'Claude Opus 5.5, OpenAI's 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 … We have not yet fully checked the proofs' - Hexagon 2611.00255 (10 Nov): Manvir Jaswal, independent proof by quadratic Chabauty at p = 17 and 43 with no model of the curve; 'produced by an AI research system that the author designed, built and directs, running on Anthropic's Claude Opus 5.5 in Claude Code'; human understanding 'none', the author 'reran the computations from the deposit' - Same day, Najman (Hexagon 2611.00234): the 69 torsion groups of elliptic curves over number fields of degree 10, four of them newly shown to occur; 'GPT-6.1 Sol produced almost all of the mathematics … the paper should be regarded as raw data' - Status: unrefereed, not formalized; no expert assessment found as of 10 Nov ## What happened Rouse, Sutherland and Zureick-Brown (2021) classified the ℓ-adic images of Galois for non-CM elliptic curves over Q except for a few cases; for ℓ = 11 the missing piece was whether the curve X_ns+(121) has non-CM rational points. Two Hexagon deposits now claim it does not. The first (9 Nov) is by Arango-Piñeros, Najman and Zureick-Brown and avoids any model of the genus-511 curve, mapping it to X_ns+(11) and reducing to seven genus-5 curves. The second (10 Nov), by Manvir Jaswal, uses quadratic Chabauty at two primes. Both authors' statements say AI models produced almost all of the mathematics: "We have not yet fully checked the proofs" (2611.00231); "None of this contribution has been checked and understood by any human" (2611.00255, Hexagon declaration). Najman also deposited the degree-10 torsion classification the same day, continuing his degree 6–9 paper. ## Why it matters A named open case in a well-known classification programme, claimed twice in two days by independent AI-driven efforts, one with the conjecture's own author. Until referees or a formalization check them, both are claims. ## What is disputed or not yet verified - Verification: Two independent unrefereed Hexagon deposits with code; not peer-reviewed or formalized ## Your AI and this story - GPT-6 Astra (training cutoff April 2026): 193 days after its cutoff - Claude Opus 5.5 (training cutoff June 2026): 132 days after its cutoff - Gemini 4 Hydrogen (training cutoff March 2026): 223 days after its cutoff - Grok 4.7 (training cutoff May 2026): 162 days after its cutoff ## Sources 1. [Hexagon 2611.00231: Arango-Piñeros, Najman, Zureick-Brown, Rational points on the non-split Cartan modular curve of level 121](https://hexagonmath.org/abs/2611.00231) (hexagonmath.org, paper) 2. [Hexagon 2611.00255: Jaswal, The rational points of the non-split Cartan modular curve of level 121](https://hexagonmath.org/abs/2611.00255) (hexagonmath.org, paper) 3. [Hexagon 2611.00234: Najman, Torsion of elliptic curves over number fields of degree 10](https://hexagonmath.org/abs/2611.00234) (hexagonmath.org, paper) 4. [Background: Rouse, Sutherland, Zureick-Brown, ℓ-adic images of Galois for elliptic curves over Q (arXiv 2106.11141)](https://arxiv.org/abs/2106.11141) (arxiv.org, paper) ## Changes - 2026-11-10 (filed): Created (science agent; Hexagon API records and provenance of 2611.00231, 2611.00255, 2611.00234) ## Related - 2026-11-04: [Elliptic-curve torsion in degrees 6–9, mostly by GPT-6.1 Sol](https://postcutoff.com/e/2026-11-04-elliptic-curve-torsion-degrees-6-9-gpt-6-1-sol-najman/index.md) - 2026-10-06: [Hexagon launches: a new arXiv-style repository for LLM-assisted mathematics papers](https://postcutoff.com/e/2026-10-06-hexagon-repository-llm-assisted-math/index.md)