## The torsion on CM elliptic curves over prime degree number fields

It’s good to be back! This weekend I’m going to Paris to give a talk in the London-Paris Number Theory seminar so I’m going to give a preview of that talk, based on joint work with Pete Clark and Abbey Bourdon. We will post this onto the arxiv soon.

Of course the paper is about the torsion in certain kinds of elliptic curves. The landmark result in this direction is of course Mazur’s theorem, stating that if $E$ is any elliptic curve over $\mathbf Q$ then the torsion subgroup of $E(\mathbf Q)$ is isomorphic to an element of the following:

$\{\mathbf Z/N\mathbf Z:N\in\{1,\ldots,10,12\}\}\cup\{\mathbf Z/2\mathbf Z\times\mathbf Z/2N\mathbf Z:N\in\{1,\ldots,4\}\}.$

Moreover each group occurs as a torsion subgroup. There is an exercise to this effect in Silverman’s Arithmetic of Elliptic Curves. Merel’s uniform boundedness theorem tells you that for any number field $F$ there is a similar but likely larger “Mazur-style list” depending only on $[F:\mathbf Q]$. This is to say for each positive integer $d$ there is a list of groups such that there is a number field $F'$ of degree $d$ and an elliptic curve $E'/F'$ with that torsion subgroup, and conversely the torsion subgroup of any elliptic curve over $F$ lies in that list.

The bad news is that when we consider all elliptic curves, it’s pretty hard to come up with these lists. A complete list is only known in degree 2 thanks to work of Sheldon Kamienny, Monsur Kenku and the late Fumiyuki Momose in the late 80s and early 90s. For higher degrees (up to 7), this is a nice accounting of the details.

The good news is that we can get a much better idea of what the torsion really is with a subset of elliptic curves. For any elliptic curve $E$ the set of maps of complex Lie groups $E(\mathbf C) \to E(\mathbf C)$ contains a copy of the integers. If it contains anything else we say that $E$ has complex multiplication or CM.

The answers here are simpler because the Galois representation of $E$ is much smaller. But at the same time, CM elliptic curves capture lots of extremal behavior. For instance, if $p>911$ is a prime, work of myself, Clark, Cook and Rice shows an instance of this for $p$-torsion points of elliptic curves over number fields.

Myself, Clark, Corn and Rice were able to compute all possible torsion subgroups over number fields of degree $\le 13$ (it was known up to degree 3 previously, please see our paper for details). Despite the fact that this was just published, this paper has been in the works for several years. We and others noticed that torsion over prime degree number fields got quite sparse.

For context, the part of Mazur’s theorem for CM elliptic curves was previously known by a theorem of Loren Olson, stating that the only possible torsion subgroups up to isomorphism over $\mathbf Q$ are what we grew to call the “Olson groups:”

$\{0,\mathbf Z/2\mathbf Z,\mathbf Z/3\mathbf Z,\mathbf Z/4\mathbf Z,\mathbf Z/6\mathbf Z,(\mathbf Z/2\mathbf Z)^2 \}.$

Schuett was one of those who noticed that for $[F:\mathbf Q] = 7,11,13$, the Olson groups were the only possible torsion of a CM elliptic curve over F. He asked if it might be that this is pattern continues for all large primes. Well, that’s exactly what myself, Bourdon and Clark just proved this past year. Moreover, over all prime degree fields there are only 17 isomorphism classes of CM elliptic curves with a torsion subgroup besides one of the Olson groups.

Now of course a new idea is needed for something like this. The key is a field $K$ called the CM field of E. The idea that we came upon is that usually if $K\not\hookrightarrow F$ and $E$ is a CM elliptic curve with $E(F)\hookleftarrow\mathbf Z/N\mathbf Z$ then $\mathbf Q(e^{2i\pi/N} + e^{-2i\pi/N})\hookrightarrow F$. Moreover if $N$ is coprime to the discriminant of the ring of maps $E \to E$ over the complex numbers, this embedding is not an isomorphism.

This is something that should be surprising! Cyclotomic fields and fields generated by CM elliptic curves are two very distinct types of extensions of the rational numbers! For instance, genus theory says that the intersection of a cyclotomic field with the field generated by the $j$-invariant of a CM elliptic curve is multiquadratic. So there’s a fundamental tension here.

We were not able to show that the CM field argument works for all fields and all $N$, but for number fields with a real embedding, we proved it. In particular, this works out whenever $[F:\mathbf Q]$ is odd, and in particular for large primes.

The point of having a real embedding means that we can use the action of complex conjugation – and we know how that acts on a CM elliptic curve over a real number field because Gauss’ genus theory ALSO tells us about the models of a CM elliptic curve over the real numbers.