Torsion of elliptic curves over number fields of degree $10$

Introduction

Let $E$ be an elliptic curve over a number field $K$. By the Mordell-Weil theorem $E(K) \simeq E(K)_{\mathrm{tors}} \times\mathbb{Z}^{r}$, and by Merel’s theorem [30] the torsion subgroup $E(K)_{\mathrm{tors}}$ is bounded in terms of $d=[K:\mathbb{Q}]$ alone. Hence, for every $d \ge1$ the set

is finite. The set $\Phi(1)$ was determined by Mazur [Maz77], building on work of Kubert [23], and $\Phi(2)$ by Kamienny [17], building on work of Kenku and Momose [22]. Derickx, Etropolski, van Hoeij, Morrow and Zureick-Brown [5] determined $\Phi(3)$, using earlier work of Jeon, Kim and Schweizer [16], Parent [40] and Najman [31]. Derickx and Najman [8] determined $\Phi(4)$, using earlier work of Jeon, Kim and Park [15] and Bruin and Najman [3]. The author determined $\Phi(5)$ in [32], and $\Phi(6)$, $\Phi(7)$, $\Phi(8)$ and $\Phi(9)$ in [35]. In this paper we determine $\Phi(10)$.

Write $\Phi^{\infty}(d) \subseteq\Phi(d)$ for the subset of groups that occur as $E(K)_{\mathrm{tors}}$ for infinitely many $\mathbb{Q}$-isomorphism classes of elliptic curves $E$ over number fields $K$ of degree $d$. The sets $\Phi^{\infty}(d)$ are known for $d \le9$: we have $\Phi^{\infty}(d)=\Phi(d)$ for $d=1,2$, the sets $\Phi^{\infty}(3)$ and $\Phi^{\infty}(4)$ were determined by Jeon, Kim and Schweizer [16] and Jeon, Kim and Park [15], $\Phi^{\infty}(5)$ and $\Phi^{\infty}(6)$ by Derickx and Sutherland [11], and $\Phi^{\infty}(7)$, $\Phi^{\infty}(8)$ and $\Phi^{\infty}(9)$ by Najman and Varivoda [NV26]. The set $\Phi^{\infty}(10)$ has not been determined, and we do not determine it here either.

The author 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.

Closely related is the set $S(d)$ of primes $p$ for which there exists an elliptic curve over a number field of degree $d$ with a point of order $p$. The sets $S(d)$ for $d \le8$ follow from the work of Mazur, Kamienny, Kenku and Momose, Parent, Derickx, Kamienny, Stein and Stoll, and Khawaja [Maz77, Kam92, KM88, Par00, DKSS23, Kha24], and Derickx and Stoll [12] developed tools that rule out most primes for any given $d$. The author [34 Theorem 1.1] recently determined $S(d)$ for $9 \le d \le22$, and in particular showed that $S(10)$ is the set of primes up to 37. This is the starting point of our proof.

Our main result is the following theorem.

Of these 69 groups, the 35 groups of $\Phi^\infty(1)\cup\Phi^\infty(2)\cup\Phi^\infty(5)$ and the 15 groups

occur for infinitely many $\overline{\mathbb{Q}}$-isomorphism classes of elliptic curves over number fields of degree 10 (Lemma 3.1 and Proposition 3.3). The groups $\mathbb{Z}/28\mathbb{Z}$, $\mathbb{Z}/30\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z}$ occur because they occur over quintic fields [32], and the groups $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/28\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/30\mathbb{Z}$ because the quintic curves with torsion $\mathbb{Z}/28\mathbb{Z}$ and $\mathbb{Z}/30\mathbb{Z}$ gain full 2-torsion over a quadratic extension (Proposition 3.5). For $N\in\{29,31,35,37,38,39,40,42,45,50\}$ the group $\mathbb{Z}/N\mathbb{Z}$ is the torsion group of an elliptic curve coming from a point of degree 10 on $X_1(N)$ found by van Hoeij [vH14] (Proposition 3.4). For the remaining 4 groups we also determine all the elliptic curves in question.

The two pairs with torsion group $\mathbb{Z}/3\mathbb{Z} \times\mathbb{Z}/18\mathbb{Z}$ are defined over the same field, and their elliptic curves are 2-isogenous. All the curves of Theorem 1.2 have $j$-invariants of degree 5 over $\mathbb{Q}$, which are not algebraic integers, so none of them has complex multiplication.

The elliptic curves of Theorem 1.2 are new, and they are listed in [33] as the points 2.26.10.a, 2.26.10.b, 3.15.10.a, 3.18.10.a, 3.18.10.b and 6.12.10.a. We found them in the course of this work, so the groups $\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/26\mathbb{Z}$, $\mathbb{Z}/3\mathbb{Z} \times\mathbb{Z}/15\mathbb{Z}$, $\mathbb{Z}/3\mathbb{Z} \times\mathbb{Z}/18\mathbb{Z}$ and $\mathbb{Z}/6\mathbb{Z} \times\mathbb{Z}/12\mathbb{Z}$ were not known to occur in degree 10. The elliptic curves with torsion groups $\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/28\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/30\mathbb{Z}$ over decic fields of Proposition 3.5 (the points 2.28.10.a, 2.30.10.a and 2.30.10.b), which we obtained from the quintic curves of van Hoeij with torsion $\mathbb{Z}/28\mathbb{Z}$ and $\mathbb{Z}/30\mathbb{Z}$, are also new.

The isomorphism of pairs is defined in § 2.2. Note that $\mathbb{Z}/37\mathbb{Z}$ occurs in degrees 6 and 10, but in none of the degrees 7, 8 and 9 [35 Theorem 1.1].

Outline of the proof and of the paper

As in [8], [32] and [35], the classification reduces to showing that certain modular curves $X_1(m,n)$ have no non-cuspidal points over number fields of degree 10. We introduce the notation and collect the general tools from [35] in § 2. In § 3 we show that the groups of Theorem 1.1 occur, apart from the 4 groups of Theorem 1.2, and in § 4 we show that 110 groups have to be ruled out. Over a number field $K$ of degree 10 with $\mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z} \subseteq E(K)$ we have $m \in\{1, 2, 3, 4, 6, 11, 22\}$, and for $m \in\{3, 4, 6\}$ the problem becomes one about points of degree 5 over $\mathbb{Q}(\zeta_m)$.

The global method of Derickx [8 Proposition 4.3], combined with arguments with diamond operators, eliminates the groups $\mathbb{Z}/n\mathbb{Z}$ of $M_{10}$ with $n \ge95$ apart from $n \in\{99, 100, 105, 117\}$ (§ 5). The groups $\mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z}$ with $m \ge3$ are treated in Section 6, and the groups $\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/2n\mathbb{Z}$, together with the cyclic groups that Lemma 4.3 reduces to them, in § 8. The remaining cyclic groups are treated in § 9 to § 11: with the Hecke sieve of [8 Proposition 5.5] and computations with the Picard groups of modular curves over finite fields, often after pushing the divisors forward to a curve of smaller level (§ 9), with formal immersion criteria at the prime 2 (§ 10), and, together with $\mathbb{Z}/3\mathbb{Z} \times\mathbb{Z}/39\mathbb{Z}$, with combinations of these methods on quotients of $X_1(n)$ (§ 11). In § 12 we construct the elliptic curves with torsion groups $\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/26\mathbb{Z}$, $\mathbb{Z}/3\mathbb{Z} \times\mathbb{Z}/15\mathbb{Z}$, $\mathbb{Z}/3\mathbb{Z} \times\mathbb{Z}/18\mathbb{Z}$ and $\mathbb{Z}/6\mathbb{Z} \times\mathbb{Z}/12\mathbb{Z}$ and determine all the points of the relevant degree on the corresponding modular curves, which also rules out the groups properly containing them. We prove Theorems 1.1 and 1.2 in § 13, and the computations are described in § 14.

Data availability and reproducibility

The code, the stored logs and the data of the computations described in this paper are available at

https://github.com/F-Najman/deg10_torsion

This paper refers to the revision 0903ee6 of this repository, and the links in the paper point to this revision. The README file there lists, statement by statement, which scripts establish each claim and which logs they produced, and also the checks that the proofs do not need (see § 14). 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

AI models were extensively used in the work leading to this paper. After completing [8] (with Derickx) and [32] and giving talks about this work, I was asked several times whether we could do such a classification for number fields of degree $d > 5$. My answer was that I thought it was not possible as it would require explicit computations that did not seem feasible to me. To check whether this was correct, in mid-October 2026 I asked AI whether this was possible, and whether it could do it. To my surprise, it was able to complete the classifications of possible torsion groups of elliptic curves over number fields, not only in degree 6, but later also in degrees 7, 8, 9 and finally 10. After some initial hesitation about releasing primarily AI-generated mathematics, I decided that the results were simply too interesting and the mathematics too beautiful to just stay on my computer.

OpenAI’s GPT-6.1 Sol produced almost all of the mathematical content of the first version of the proof and carried out most of the computations. Anthropic’s Claude Opus 5.5 reviewed that version, independently recomputed some of the computations (see § 14) and the explicit examples, wrote the draft of the present manuscript and organized the code repository.

Notation and auxiliary results

We will be using the conventions and notation from [35 Section 2], which we will essentially repeat here for the convenience of the reader. We also collect the auxiliary results of [35] that we use. They are stated there for number fields of degree 6, 7, 8 and 9, but most of them are stated for an arbitrary degree $d$, and their proofs do not use the degree otherwise.

Modular curves