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The rational points of the non-split Cartan modular curve of level 121

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by Manvir Jaswal ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

We prove that the modular curve Xns+(121)X_{\mathrm{ns}}^{+}(121) of the normalizer of a non-split Cartan subgroup of level 121, of genus 511, has exactly seven rational points, all of them CM. This is the level 121 case of a conjecture of Rouse, Sutherland and Zureick-Brown, and with their theorem it determines the 11-adic images of Galois of non-CM elliptic curves over Q\mathbb{Q}. The Jacobian has rank equal to the genus. We work at the primes 17 and 43, at each of which one Hecke block of the Mordell–Weil group has logarithm divisible by p2p^2 while its pp-adic height has valuation exactly one. A comparison of valuations in the quadratic Chabauty identity then shows that the residue disc of each CM point holds no other rational point, with no computation of the local height at pp, no basis of the Mordell–Weil group and no model of the curve. The inputs come from expansions of differentials at CM points and from a formula we prove for intersection numbers of CM divisors under Hecke correspondences on Xns+(ℓe)X_{\mathrm{ns}}^{+}(\ell^e), at every prime v≠ℓv \ne \ell, and at v=ℓv = \ell when ℓ≥7\ell \ge 7, ℓ≡3 mod 4\ell \equiv 3 \bmod 4 and e≤2e \le 2.

Provenance statement

This paper and its results were produced by an AI research system that the author designed, built and directs, running on Anthropic's Claude Opus 5.5 in Claude Code. The author chooses the problems and prioritizes the work. Separate AI agents checked each result, and a second program confirmed the computations the proofs use, with any exceptions listed in the public deposit. The system also searched the literature and checked the citations against reliable sources. The author read the paper, reran the computations from the deposit and takes full responsibility for all of the paper's contents.

Tools used

Anthropic
Claude OpusVersion 5.5
Anthropic
Claude Code
SageMath
SageMathVersion 10.9
PARI/GP
PARI/GPVersion 2.17.3

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Version history

  1. v1 · 40 pagesSubmitted by Manvir JaswalInitial depositCurrentNov 10, 2026