Classification of torsion of elliptic curves over quintic fields

Dedicated with gratitude and admiration to my advisor Andrej Dujella on the occasion of his 60th birthday.

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 [26] the torsion subgroup $E(K)_{\mathrm{tors}}$ is bounded in terms of $d = [K : \mathbb{Q}]$ alone. Hence, for every $d \geq1$ the set

is finite. Determining $\Phi(d)$ has been one of the central problems in the arithmetic of elliptic curves. The set $\Phi(1)$ was determined by Mazur [24], building on work of Kubert [21]; $\Phi(2)$ by Kamienny [16], building on work of Kenku and Momose [20]; $\Phi(3)$ by Derickx, Etropolski, van Hoeij, Morrow and Zureick-Brown [6], building on work of Jeon, Kim and Schweizer [15], Parent [28] and Najman [27]; and $\Phi(4)$ by Derickx and Najman [8], building on work of Jeon, Kim and Park [14] and Bruin and Najman [4]. In this paper we determine $\Phi(5)$.

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 $d$. A group in $\Phi(d) \setminus\Phi^{\infty}(d)$ will be called a sporadic torsion group in degree $d$. Following [8], a point of degree $d$ on a curve $X$ is called sporadic if $X$ has only finitely many points of degree at most $d$; this is a property of a point on a modular curve, not of a group. One has $\Phi^{\infty}(d) = \Phi(d)$ for $d = 1, 2$; there is exactly one sporadic torsion group in degree 3, namely $\mathbb{Z}/21\mathbb{Z}$, which is realised by a single elliptic curve, the base change to $\mathbb{Q}(\zeta_{9})^{+}$ of the curve $162.c3$ [27, 6]; and there are no sporadic torsion groups in degree 4 [8]. The set $\Phi^{\infty}(5)$ was determined by Derickx and Sutherland [11], Theorem 1.1, the cyclic part being due to Derickx and van Hoeij [12], Theorem 3:

Our main result is the following.

Theorem 1.1. Let $K$ be a number field of degree 5 and let $E$ be an elliptic curve over $K$. Then $E(K)_{\mathrm{tors}}$ is isomorphic to one of the following 35 groups:

Equivalently, $\Phi(5)=\Phi^{\infty}(5)\cup\{\mathbb{Z}/28\mathbb{Z},\mathbb{Z}/30\mathbb{Z},\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z}\}$. Each of the groups in $\Phi^{\infty}(5)$ is the torsion group of infinitely many pairs $(K,E)$, whereas for the three exceptional groups the numbers of isomorphism classes of pairs $(K,E)$ are 1, 2 and 1, respectively. The corresponding curves have quintic $j$-invariants and therefore give 5, 10 and 5 $\mathbb{Q}$-isomorphism classes, respectively. They are listed explicitly in Theorem 1.2.

Thus $d=5$ is the first degree in which more than one sporadic torsion group occurs, and $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z}$ is the first non-cyclic sporadic torsion group in any of the degrees $d\le5$ for which $\Phi(d)$ is known. The three sporadic groups are described completely by the following theorem, which is proved in Section 9.

Theorem 1.2. Let $K$ be a number field of degree 5 and let $E/K$ be an elliptic curve such that $E(K)_{\mathrm{tors}}$ is not in $\Phi^{\infty}(5)$. Then, up to isomorphism of the pair $(K,E)$, exactly one of the following holds.

(1) $E(K)_{\mathrm{tors}}\cong\mathbb{Z}/28\mathbb{Z}$, $K=\mathbb{Q}(\alpha)$ with $\alpha^{5}-\alpha^{4}-2\alpha^{3}-\alpha^{2}+2\alpha+2=0$, and $E$ is the curve (6) in Section 9.

(2) $E(K)_{\mathrm{tors}}\cong\mathbb{Z}/30\mathbb{Z}$, $K=\mathbb{Q}(\alpha)$ with $\alpha^{5}+\alpha^{4}-3\alpha^{3}+3\alpha+1=0$, and $E$ is one of the two curves (7), (8) in Section 9, which are 2-isogenous over $K$.

(3) $E(K)_{\mathrm{tors}}\cong\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z}$, $K=\mathbb{Q}(w)$ with $w^{5}-2w^{4}+2w^{3}-3w^{2}-w+4=0$, and $E$ is the curve (9) in Section 9.

In all three cases $K$ has signature $(3,1)$ and class number 1 and the Galois group of its normal closure is $S_{5}$; each of the four curves is without complex multiplication and has a $j$-invariant of degree 5 over $\mathbb{Q}$. The discriminants, LMFDB labels and conductors are displayed in Table 7.

$E(K)_{\mathrm{tors}}$	$E$	$K$ (LMFDB label)	$\operatorname{disc} K$	$\mathrm{N}f_E$
$\mathbb{Z}/28\mathbb{Z}$	$E_{28}$	5.3.17348.1	$-2^2 \cdot4337$	16
$\mathbb{Z}/30\mathbb{Z}$	$E_{30}$	5.3.10407.1	$-3 \cdot3469$	33
$\mathbb{Z}/30\mathbb{Z}$	$E'_{30}$	5.3.10407.1	$-3 \cdot3469$	33
$\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/18\mathbb{Z}$	$E_{2,18}$	5.3.34779.1	$-3 \cdot11593$	6

The curves in cases (1) and (2) were found by van Hoeij [31]; what is new here is that his list is complete, and that the corresponding torsion groups are exactly $\mathbb{Z}/28\mathbb{Z}$ and $\mathbb{Z}/30\mathbb{Z}$. The curve in case (3) is new. In the course of the proof we determine all quintic points on the three relevant modular curves.

Theorem 1.3. (1) $X_1(28)$ has exactly 6 non-cuspidal closed points of degree 5; they form a single free orbit under the group $\Delta(28)$ of diamond operators, and correspond to the six pairs $\{P,-P\}$ of points $P$ of order 28 on the curve $E$ of Theorem 1.2(1).

(2) $X_1(30)$ has exactly 8 non-cuspidal closed points of degree 5; they form two free orbits under $\Delta(30)$, and correspond to the four pairs $\{P,-P\}$ of points $P$ of order 30 on each of the two curves of Theorem 1.2(2).

(3) $X_1(2,18)$ has exactly 18 non-cuspidal closed points of degree 5; they correspond to the 18 equivalence classes, modulo $\pm1$, of embeddings $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z}\hookrightarrow E(K)$, for the curve $E$ of Theorem 1.2(3).

In each case the residue field of every one of these points is the corresponding quintic field $K$ of Theorem 1.2.

Theorem 1.1 also settles, for every level at once, the question of which modular curves of the two families $X_1(n)$ and $X_1(2,2n)$ carry a point of degree 5.

Corollary 1.4. (1) $X_1(n)$ has a non-cuspidal point of degree 5 if and only if $n \in\{1,\ldots,22,24,25,28,30\}$. For $n \le25$, $n \ne23$, there are infinitely many such points; for $n=28$ there are exactly 6 and for $n=30$ exactly 8.

(2) $X_1(2,2n)$ has a non-cuspidal point of degree 5 if and only if $n \le9$. For $n \le8$ there are infinitely many such points, and for $n=9$ there are exactly 18.

(3) Consequently $X_1(28)$ and $X_1(30)$ are the only curves in these two families that have a non-cuspidal sporadic point of degree 5.

The proof of Corollary 1.4 is given in Section 9, together with the reason why the word non-cuspidal cannot be dropped from part (3): the curve $X_1(44)$ has a cuspidal sporadic point of degree 5. The three exceptional groups behave differently as far as the modular curves are concerned. For $\mathbb{Z}/28\mathbb{Z}$ and $\mathbb{Z}/30\mathbb{Z}$ the corresponding points are sporadic points of $X_1(28)$ and $X_1(30)$; for $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z}$ they are not, because $\operatorname{gon}_{\mathbb{Q}} X_1(2,18)=4$ and this curve has infinitely many quartic points. They are nevertheless isolated in the sense of Bourdon, Ejder, Liu, Odumodu and Viray [3], Definition 4.1. The example has a feature not seen in lower degrees: by [14], Theorem 3.6 we have $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z}\in\Phi^\infty(4)$, so $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z}$ is the torsion group of infinitely many elliptic curves over quartic fields, and over quintic fields of a unique pair $(K,E)$ up to isomorphism. See Section 9.

Outline of the proof and of the paper

As in [8] and previous works, the classification reduces to showing that certain modular curves $X_1(m,n)$ have no non-cuspidal points of degree 5. In Section 3 we recall why exactly 45 groups have to be analysed. Nineteen of them were eliminated, for quintic fields, in [8], §7, and four more, namely $\mathbb{Z}/4n\mathbb{Z}$ for $9\le n\le12$, were reduced there to the four groups $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2n\mathbb{Z}$. We recall that argument in Section 4, with the numerical conditions checked for $d=5$ rather than $d=4$; one point of formulation, which is precisely what puts $\mathbb{Z}/77\mathbb{Z}$ and $\mathbb{Z}/85\mathbb{Z}$ outside its reach in degree 5, is discussed in Remark 4.3.

The twenty-two groups that [8] does not reach are treated here; $\mathbb{Z}/36\mathbb{Z}$ then needs one further argument, because $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z}$ turns out to occur. The main tools are: