Rational points on the non-split Cartan modular curve of level $121$

Introduction

Let $E$ be an elliptic curve over $\mathbb{Q}$ without complex multiplication (CM) and let $\ell$ be a prime. Serre [14] proved that the image of the $\ell$-adic Galois representation $\rho_{E,\ell^\infty}: G_{\mathbb{Q}} \to\operatorname{GL}_{2}(\mathbb{Z}_{\ell})$ is open, and that the mod $\ell$ representation $\overline{\rho}_{E,\ell}$ is surjective for all but finitely many $\ell$. He asked whether the primes $\ell$ for which $\overline{\rho}_{E,\ell}$ is not surjective can be bounded independently of $E$, and it is expected that over $\mathbb{Q}$ the bound 37 suffices (see [18]). For $\ell> 37$, by the work of Mazur [8], Serre [14] and Bilu, Parent and Rebolledo [3], if $\overline{\rho}_{E,\ell}$ is not surjective, then its image is contained in the normalizer of a non-split Cartan subgroup of $\operatorname{GL}_{2}(\mathbb{F}_{\ell})$, so this question becomes the question whether the modular curves $X_{\mathrm{ns}}^{+}(\ell)$ have non-CM rational points.

Rouse, Sutherland and Zureick-Brown [10] determined, for every prime $\ell$, the possible $\ell$-adic images of non-CM elliptic curves over $\mathbb{Q}$, apart from the images contained in the normalizer of a non-split Cartan subgroup of level $\ell\ge19$ and apart from the images corresponding to rational points on finitely many explicit modular curves, whose rational points they could not determine: the curves $X_{\mathrm{ns}}^{+}(N)$ for $N=27,25,49,121$, and two curves of level 49 and genus 9 (see Theorem 1.6 of [10]). Balakrishnan, Betts, Hast, Jha and Müller [2] have since determined the rational points on $X_{\mathrm{ns}}^{+}(27)$, Furio and Lombardo [6] the rational points on $X_{\mathrm{ns}}^{+}(49)$, and Arango-Piñeros and Zureick-Brown [1] showed that the two curves of level 49 and genus 9 have no non-CM rational points. The curve $X_{\mathrm{ns}}^{+}(121)$ has genus 511, and Rouse, Sutherland and Zureick-Brown did not compute a model of it. Using the map $\varphi:X_{\mathrm{ns}}^{+}(121) \to X_{\mathrm{ns}}^{+}(11)$ to an elliptic curve of rank 1, they showed with a Mordell-Weil sieve that a non-CM rational point on $X_{\mathrm{ns}}^{+}(121)$ would map to a multiple

2020 Mathematics Subject Classification. 11G05, 11G18, 14G05, 11F80.

Key words and phrases. modular curves, non-split Cartan subgroups, Galois representations, Selmer groups, Chabauty method.

F.N. was supported by the Croatian Science Foundation under the project no. IP-2022-10-5008, by the project “Implementation of cutting-edge research and its application as part of the Scientific Center of Excellence for Quantum and Complex Systems, and Representations of Lie Algebras”, PK.1.1.10.0004, co-financed by the European Union through the European Regional Development Fund – Competitiveness and Cohesion Programme 2021–2027, and by the European Union – NextGenerationEU through the National Recovery and Resilience Plan 2021–2026, via an institutional grant of the University of Zagreb Faculty of Science, IK IA 1.1.3, Impact4Math. $kP$ of a generator $P$ with $\lvert k\rvert\ge\operatorname{lcm}(1,2,\ldots,233)$, and hence its $j$-invariant would have height roughly $10^{7.92\cdot10^{202}}$ (see [10 Section 9.2]).

In this paper we determine the rational points on $X^{+}_{\mathrm{ns}}(121)$. Our main result is the following theorem.

Note that these are all the discriminants of imaginary quadratic orders of class number 1 in which 11 is inert, so the rational points on $X^{+}_{\mathrm{ns}}(121)$ are the ones that the theory of complex multiplication predicts. We obtain the following corollary.

Proof. If $\rho_{E,121}(G_{\mathbb{Q}})$ were contained in such a group, then $j(E)$ would be the $j$-invariant of a non-CM rational point on $X^{+}_{\mathrm{ns}}(121)$, contradicting Theorem 1.1. The second statement now follows from [10 Theorem 1.6], as for $\ell=11$ we have just excluded case (iii) of that theorem, and case (iv) occurs only for $\ell=7$. □

Thus the 11-adic images of Galois of non-CM elliptic curves over $\mathbb{Q}$ are now completely determined. Of the curves listed in [10 Theorem 1.6], this leaves only $X^{+}_{\mathrm{ns}}(25)$. While writing this paper, we became aware that Eray Karabıyık has determined the rational points on $X^{+}_{\mathrm{ns}}(25)$ (private communication).

We now describe our methods. We never use a model of $X^{+}_{\mathrm{ns}}(121)$. The only information about $X^{+}_{\mathrm{ns}}(121)$ that we use is the moduli interpretation of the map $\varphi: X^{+}_{\mathrm{ns}}(121) \to X^{+}_{\mathrm{ns}}(11)$ of degree 121, at the cusps and at the CM points. The curve $X^{+}_{\mathrm{ns}}(11)$ is the elliptic curve

of rank 1, with $E(\mathbb{Q})=\mathbb{Z}G$, where $G=(4,5)$. Its 5 cusps are defined over $K=\mathbb{Q}(\zeta_{11})^{+}$, the maximal real subfield of $\mathbb{Q}(\zeta_{11})$, and they are points of order 11 on $E$. The proof has three steps.

The first step is a descent with the cusps. If $\xi\in X^{+}_{\mathrm{ns}}(121)(\mathbb{Q})$, then for every prime $q$ of $K$ not above 11 and every cusp $c$ of $X^{+}_{\mathrm{ns}}(11)$, the point $\varphi(\xi) \in E(\mathbb{Q})$ is congruent to $c$ modulo $q^{i}$ only if it is congruent to $c$ modulo $q^{11\lceil i/11\rceil}$, i.e. the intersection number of $\varphi(\xi)$ and $c$ at $q$ is divisible by 11 (Lemmas 4.2 and 4.3). This is the analogue for $X^{+}_{\mathrm{ns}}(121)$ of the condition, used by Schoof and Tzanakis [13] to determine the points on $X^{+}_{\mathrm{ns}}(11)$ with integral $j$-invariant, that $\varphi(\xi)$ does not meet the cusps at all. It follows that certain functions on $E$ whose divisors are supported on the cusps and the origin take values in the finite group $K(S,11)$, where $S$ consists of the prime of $K$ above 11. A Mordell-Weil sieve with 6 primes then shows that $\varphi(\xi)=nG$ with $n \equiv n_{0} \pmod{4356}$ for some $n_{0} \in\{-2,\ldots,4\}$, and determines the values of these functions modulo 11-th powers (Proposition 4.7). The 7 values of $n_{0}$ correspond to the 7 CM points, and since 121 divides 4356, it follows that $n-n_{0}$ is divisible by 121.

The second step reduces the problem to curves of genus 5 (§ 5). The values from the first step show that $x(nG-3s)=\alpha+\delta_{n_{0}}w^{11}$ with $w \in K$, where $s \in E(K)$ is a cusp and $\alpha,\delta_{n_{0}} \in K$ are explicit. Combined with a 2-descent on $E$ over a cubic extension $L$ of $K$, this gives a point on one of 7 hyperelliptic curves $\eta y^{2}=x^{11}+b^{2}$ of genus 5 over $L$, which is a number field of degree 15. The CM point gives the point with $w=1$.

The third step deals with these curves. Over $F=L(\zeta_{11})$ their Jacobians $J$ have the automorphism $\sigma$ induced by $X \mapsto\zeta_{11}X$, and $\lambda=1-\sigma$ is an isogeny of degree 11. We use the idea of Stoll’s Selmer group Chabauty [16 Theorem 2.1 and Remark 2.2] for the isogeny $\lambda$. Since 121 divides $n-n_0$, the point $[P-P_0] \in J(F)$, where $P$ is the point coming from $\xi$ and $P_0$ the point coming from the CM point, is deep in the kernel of reduction at a prime $v$ above 11. As long as a point is deep enough, its image under the local $\lambda$-Kummer map is 0, and since the localization at $v$ of the $\lambda$-Selmer group turns out to be injective, the point is divisible by $\lambda$ in $J(F)$. Repeating this, we arrive at a point of the kernel of reduction whose leading term we know. Its image under the local Kummer map is not in the image of the Selmer group, which is a contradiction (Propositions 8.3 and 9.2).

The method works because the curves $Y^2=X^{11}+b^2$ have cuspidal reduction $Y^2=(X+\bar b^2)^{11}$ at the primes above 11. As a consequence, the formal groups of their Jacobians at these primes are isomorphic to $\widehat{\mathbb{G}}_a^5$ via the formal logarithm (Lemma 6.1), and $\lambda$ acts diagonally in the logarithmic coordinates, so the repeated divisions by $\lambda$ can be followed explicitly. Note that our approach needs neither generators of the Mordell–Weil groups of the Jacobians (which have rank 5 over $L$ for 3 of the 7 curves, see Remark 5.3) nor a bound on the height of the points. We do not assume the Generalized Riemann Hypothesis (GRH). It is used only to find generators of groups of $S$-units, which are then shown to be correct unconditionally.

We fix our notation in Section 2 and prove Theorem 1.1 in Section 10. The computations are described in Section 11.

All the code used in the paper can be found at

https://github.com/F-Najman/Xns-121-

Data availability and reproducibility

The repository contains the code, the logs and the certificates of all the computations in this paper, and a script that reruns all of them (in about 4 minutes). This paper refers to the revision c12c8c3 of the repository, and the links in the paper point to this revision. The README file there lists, statement by statement, which script establishes each claim and which log it produced. In the digital form of this article, the words reporting a computation, such as “we compute”, are hyperlinks to the script that carries it out.

AI disclosure and comments

The mathematical content of this paper was produced by AI models. Anthropic’s 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. Claude Opus 5.5 also reviewed these proofs, reran all the computations, independently reimplemented the sieve, the Selmer group computations for the even classes, the determination of the rational points above the CM points and the computation of the genus, checked the references, wrote the manuscript and organized the code repository. We have not yet fully checked the proofs.

Notation and conventions

Throughout the paper $\zeta_{11}$ denotes a primitive 11-th root of unity, and $\pi=1-\zeta_{11}$. For a number field $k$, a finite set $S$ of primes of $k$ and a prime $p$ we write

This is a finite dimensional $\mathbb{F}_p$-vector space, which we write additively. If $k$ contains a primitive $p$-th root of unity and $\operatorname{Cl}(k)[p]=0$, then $\dim_{\mathbb{F}_p}k(S,p)=r_1+r_2+\#S$, where $(r_1,r_2)$ is the signature of $k$. For a finite extension $l$ of $\mathbb{Q}_p$ we denote by $v$ or $v_l$ the valuation of $l$ normalized by $v(l^\times)=\mathbb{Z}$, and by $\mathcal{O}_l$ its ring of integers.

Let $\varepsilon=2$, which is not a square modulo 11. The non-split Cartan subgroup $C_{\mathrm{ns}}(121)$ of $\mathrm{GL}_2(\mathbb{Z}/121\mathbb{Z})$ is the group of invertible matrices of the form