Torsion for CM Elliptic Curves Defined over Number Fields of Degree $2p$

In 1922, Mordell proved that the set of $\mathbb{Q}$-rational points of an elliptic curve $E$ defined over $\mathbb{Q}$ is a finitely generated abelian group [23]. That is, $E(\mathbb{Q})\cong E(\mathbb{Q})_{\operatorname{tors}}\times\mathbb{Z}^{r}$, where $E(\mathbb{Q})_{\operatorname{tors}}$ denotes the finite set of torsion points and $r\in\mathbb{Z}^{\geq 0}$ is the rank of $E/\mathbb{Q}$. It is natural to ask what groups arise as $E(\mathbb{Q})_{\operatorname{tors}}$ as $E$ ranges over all elliptic curves over $\mathbb{Q}$, and the answer is known due to work of Mazur.

More generally, if $E$ is an elliptic curve defined over a number field $F$, then the set of $F$-rational points of $E$ is again a finitely generated abelian group by Weil [31], so one may seek to classify the groups occurring as $E(F)_{\operatorname{tors}}$. In fact, by Merel’s Uniform Boundedness Theorem [22], there are only finitely many groups that arise as $E(F)_{\operatorname{tors}}$, even as $E$ ranges over all elliptic curves defined over all number fields $F$ of a fixed degree. Thus the fundamental question which motivates our work is the following:

Now 100 years after Mordell’s proof, the answer to Question 1 is known only for $d\leq 3$; see [21, 18, 20, 19, 11]. A fundamental obstruction to extending the classification to $d>3$ is the existence of so-called sporadic or isolated points on modular curves which can give rise to torsion subgroups occurring on only finitely many elliptic curves (up to isomorphism) defined over all number fields of a fixed degree. To date, we lack adequate tools for detecting such points, and hence the problem of classifying torsion subgroups of elliptic curves over higher degree number fields remains largely open.

One way to obtain classification results beyond cubic fields is to restrict the elliptic curves under consideration. One common family of elliptic curves to study in this context is elliptic curves $E/\mathbb{Q}$ under base extension, where the classification of torsion subgroups is known for degrees $d\leq 5,$ $d=7$, or $d$ not divisible by a prime $\leq 7$; see [24, 13, 12]. If we require only that the $j$-invariant of $E$ lie in $\mathbb{Q}$, then analogous classification results exist [14, 10]. Another common family is elliptic curves with complex multiplication (CM), which are elliptic curves with unusually large endomorphism rings. Whereas most elliptic curves have endomorphism ring isomorphic to $\mathbb{Z}$, we say $E/F$ is a CM elliptic curve if $\operatorname{End}_{\overline{F}}(E)\cong\mathcal{O}$, an order in an imaginary quadratic field $K$. Each order is uniquely determined by its discriminant $\Delta\coloneqq[\mathcal{O}_{K}:\mathcal{O}]^{2}\cdot\Delta_{K}$, where $\Delta_{K}$ is the discriminant of $K$ and $\mathcal{O}_{K}$ is its ring of integers. For the set of all CM elliptic curves, the classification of torsion subgroups is known for any $d\leq 13$ or for any odd $d>13$; see [6, 25, 5, 4]. We note that CM elliptic curves produce many examples of sporadic points on modular curves (see, for example, [7]), so this provides further motivation for studying this class in particular.

In the present work, we extend the classification of torsion subgroups of CM elliptic curves to those defined over any number field of degree twice a prime, building on work of the first author and Clark [1, 2]. In fact, since the classification is known for $d=4,6,$ and 10 by [6], we need only consider fields of degree $2p$ for primes $p>5$. Our classification is most clearly stated in the context of new subgroups. By Theorem 2.1 in [4], if a torsion subgroup arises in degree $d^{\prime},$ then it arises in any degree $d$ for which $d^{\prime}\mid d.$ We say a CM torsion subgroup is new if it occurs in degree $d$ and not in any degree $d^{\prime}<d$ such that $d^{\prime}\mid d.$ Since torsion subgroups of CM elliptic curves in degrees 1 and 2 are known [25, 6], and there are no new CM torsion subgroups in degree $p>5$ for $p$ prime [4], it suffices to classify only the new subgroups arising in degree $2p$.

Theorem 1.2 tells us that if $\Delta\neq-3,-4$, then the only $\Delta$-CM torsion subgroups that can arise in degree $2p$ for $p>5$ that did not occur over a number field of degree 2 or degree $p$ must have exponent $2p+1$ or $2(2p+1)$, where $p$ is a Sophie Germain prime. It is conjectured that there are infinitely many Sophie Germain primes, though this remains unproven. These primes were a vital piece of Sophie Germain’s investigations concerning Fermat’s Last Theorem.

From Theorem 1.2, we can quickly deduce the torsion subgroups that arise for CM elliptic curves defined over number fields of degree $2p$ where $p>5$ is prime, including for the first previously unknown degree $d=14$. For example, 7 is not a Sophie Germain prime, but $6\cdot 7+1$ and $4\cdot 7+1$ are both prime. Thus, by Theorem 1.2, the new torsion subgroups in degree 14 are precisely $\mathbb{Z}/43\mathbb{Z},$ $\mathbb{Z}/49\mathbb{Z},$ and $\mathbb{Z}/58\mathbb{Z}$. We record this and other small degrees in the following result.

In particular, by Corollary 1.4, we see that no new torsion subgroups arise on CM elliptic curves defined over number fields of degree $d=2\cdot 19$. Thus, another consequence of Theorem 1.2 is a description of degrees of the form $2p$ such that no new torsion subgroups occur.

This finding is significant in the context of “stratification of torsion,” a phenomenon first explored in [3, 5] for CM torsion subgroups in odd degree. For any positive integer $d$, let $\mathscr{G}_{\operatorname{CM}}(d)$ denote the set of isomorphism classes of groups which arise as $E(F)_{\operatorname{tors}}$ for some CM elliptic curve $E$ over some degree $d$ number field $F$. For any positive integer $d$, we define the set of $d$-Olson degrees to be those positive integers $d^{\prime}$ for which $\mathscr{G}_{\operatorname{CM}}(d^{\prime})=\mathscr{G}_{\operatorname{CM}}(d)$. In the case of odd $d$, we find that the set of $d$-Olson degrees possesses a positive asymptotic density [5], but whether the same holds true for any even $d$ is still an open problem. See [5, Questions 1.6].

We thank Frank Moore, Jeremy Rouse, and the anonymous referees for helpful comments on an earlier draft, and we thank Pete L. Clark for Remark 1.7 and other helpful comments. The first author was partially supported by an A.J. Sterge Faculty Fellowship and NSF grant DMS-2137659.

For most elliptic curves $E$ over a number field $F$, the ring of endomorphisms of $E$ defined over $\overline{F}$ is isomorphic to $\mathbb{Z}$, where $n\in\mathbb{Z}$ corresponds to the multiplication-by-$n$ map on $E$. We say an elliptic curve has complex multiplication, or CM, if its endomorphism ring is strictly larger than $\mathbb{Z}.$ For a CM elliptic curve $E/F$, there is an imaginary quadratic field $K$ and positive integer $f$ such that End${}_{\bar{F}}(E)\cong\mathcal{O}=\mathbb{Z}+f\mathcal{O}_{K},$ the order in $K$ of conductor $f$. Here $\mathcal{O}_{K}$ denotes the full ring of integers in $K$. In particular, we note that $\mathcal{O}\subseteq\mathcal{O}_{K}$ and $[\mathcal{O}_{K}:\mathcal{O}]=f$. The order is largest when $f=1$, and so we call $\mathcal{O}_{K}$ the maximal order. Any order $\mathcal{O}$ in an imaginary quadratic field $K$ can be uniquely identified using its discriminant,

where $\Delta_{K}$ is the discriminant of $K$. We let $\omega$ denote the number of units in $\mathcal{O}$, so

For an elliptic curve $E$ with CM by the order of discriminant $\Delta$, we have $\Delta=-3$ if and only if $j(E)=0$ and $\Delta=-4$ if and only if $j(E)=1728$. We use $w_{K}$ to denote $\#\mathcal{O}_{K}^{\times}$.

CM elliptic curves have a well-known and beautiful connection with class field theory. For example, if $E$ has CM by the maximal order in $K$, then $K(j(E),\mathfrak{h}(E_{\operatorname{tors}}))$ is the maximal abelian extension of $K$, where $\mathfrak{h}:E\rightarrow E/\operatorname{Aut}(E)\cong\mathbb{P}^{1}$ denotes a Weber function on $E$. If one adjoins the values of a Weber function only on points of order dividing $N$, we obtain the ray class field of $K$ modulo $N$; see, for example Theorem II.5.6 and Corollary II.5.7 of [30]. Of particular relevance to the present work is the fact that if $E$ has CM by the order in $K$ of conductor $f$, then $K(j(E))$ is the ring class field of $K$ of conductor $f$ and $[K(j(E)):K]=[\mathbb{Q}(j(E)):\mathbb{Q}]=h(\mathcal{O})$, the class number of $\mathcal{O}$. For an elliptic curve $E$ with CM by the order of discriminant $f^{2}\Delta_{K}$ with $f\geq 2$, we have [9, Cor. 7.24]

where $h_{K}$ denotes the class number of $K$ and $\left(\frac{\Delta_{K}}{p}\right)$ is the Kronecker symbol. As there are only finitely many imaginary quadratic fields of a given class number [17, Theorem III], there are only finitely many imaginary quadratic orders of a given class number by (1). For each imaginary quadratic order $\mathcal{O}$, there are precisely $h(\mathcal{O})$ non-isomorphic $\mathcal{O}$-CM elliptic curves.

A crucial ingredient in the proof of our main result is the following theorem. Recall $\omega=\#\mathcal{O}^{\times}$.

From this, we deduce the following corollary, which also appears as Theorem 6.2 in [1]. It refines earlier results of Silverberg [28, 29].

Suppose $E/F$ is an $\mathcal{O}$-CM elliptic curve with $\mathbb{Z}/M\mathbb{Z}\times\mathbb{Z}/N\mathbb{Z}\hookrightarrow E(F)$. Since $[K(j(E)):K]=[\mathbb{Q}(j(E)):\mathbb{Q}]=h(\mathcal{O})$, we can actually consider the divisibility conditions in Theorem 2.1 over $\mathbb{Q}(j(E))$, as illustrated in the field diagram below.

Following [2], for any imaginary quadratic order $\mathcal{O}$ and integers $M\mid N$, we let $T^{\circ}(\mathcal{O},M,N)$ denote the least degree of an extension $F/\mathbb{Q}(j(E))$ in which an $\mathcal{O}$-CM elliptic curve $E/F$ has $\mathbb{Z}/M\mathbb{Z}\times\mathbb{Z}/N\mathbb{Z}\hookrightarrow E(F)$. In particular, $F$ need not contain the CM field $K$. We note $T^{\circ}(\mathcal{O},M,N)=2^{\epsilon}\cdot T(\mathcal{O},M,N)$, where $\epsilon\in\{0,1\}$. Explicit formulas for $T^{\circ}(\mathcal{O},M,N)$ for fixed $\mathcal{O}$ are computed in [2, $\S 8$].

In the case where $M=1$, we use the streamlined notation $T(\mathcal{O},N)\coloneqq T(\mathcal{O},1,N)$ and $T^{\circ}(\mathcal{O},N)\coloneqq T^{\circ}(\mathcal{O},1,N)$. We have the following description of $T^{\circ}(\mathcal{O},N)$, which follows from Theorems 1.3, 6.1, 6.2, and 6.6 in [2].

Let $E/F$ be an $\mathcal{O}$-CM elliptic curve and $P\in E$ a point of order $N$. If $[F:\mathbb{Q}]=T^{\circ}(\mathcal{O},N)\cdot h(\mathcal{O})$, then $F=\mathbb{Q}(j(E),\mathfrak{h}(P))$, where $\mathfrak{h}:E\rightarrow E/\operatorname{Aut}(E)\cong\mathbb{P}^{1}$ is a Weber function on $E$. Moreover, if $\psi:E\rightarrow E^{\prime}$ is an isomorphism, then $\mathfrak{h}(P)=\mathfrak{h}(\psi(P))$ by [27, p.107]. It follows that for any $P\in E$, the fields $K(j(E),\mathfrak{h}(P))$ and $\mathbb{Q}(j(E),\mathfrak{h}(P))$ do not depend on the chosen Weierstrass equation for $E$. See [2, $\S 2.4$] and [1, $\S 7A$] for additional details.

Let $p>5$ be a prime number, and suppose $F$ is a number field of degree $2p$. Let $E/F$ be a CM elliptic curve with $E(F)_{\operatorname{tors}}\cong\mathbb{Z}/M\mathbb{Z}\times\mathbb{Z}/N\mathbb{Z}$ for $M\mid N$. By definition, this torsion subgroup is new if it does not occur as the torsion subgroup of a CM elliptic curve defined over a number field of degree $1$, $2$, or $p$. However, every CM torsion subgroup arising in degree 1 also arises in degree 2, and there are no new torsion subgroups of CM elliptic curves in prime degree $p>5$ by [4, Theorem 1.4]. Thus $E(F)_{\operatorname{tors}}$ is new if and only if it does not occur in degree 2.

In this section, we will determine the possible exponents of a new CM torsion subgroup in degree $2p$. If $E(F)_{\operatorname{tors}}\cong\mathbb{Z}/M\mathbb{Z}\times\mathbb{Z}/N\mathbb{Z}$ is a new torsion subgroup, then either $N$ appears already as the exponent of a CM torsion subgroup in degree 2 and $N\in\{1,2,3,4,6,7,10\}$ by [6, $\S 4.2$], or else it has exponent outside this list. We say $E(F)_{\operatorname{tors}}$ has a new exponent $N$ if $E(F)_{\operatorname{tors}}$ is new and $N\not\in\{1,2,3,4,6,7,10\}$.