Torsion Subgroups of CM Elliptic Curves over Odd Degree Number Fields:

Abstract

Let |$\mathscr{G}_{\operatorname{CM}}(d)$| denote the collection of groups (up to isomorphism) that appear as the torsion subgroup of a complex multiplication (CM) elliptic curve over a degree |$d$| number field. We completely determine |$\mathscr{G}_{\operatorname{CM}}(d)$| for odd integers |$d$| and deduce a number of statistical theorems about the behavior of torsion subgroups of CM elliptic curves. Here are three examples: (1) For each odd |$d$|⁠, the set of natural numbers |$d'$ with $\mathscr{G}_{\operatorname{CM}}(d')=\mathscr{G}_{\operatorname{CM}}(d)$| possesses a well-defined, positive asymptotic density. (2) Let |$T_{\operatorname{CM}}(d) = \max_{G \in \mathscr{G}_{\operatorname{CM}}(d)} \#G$|⁠; under the Generalized Riemann Hypothesis, (12eγπ)2/3≤lim supd→∞d oddTCM(d)(dlog⁡log⁡d)2/3≤(24eγπ)2/3. (3) For each |$\epsilon > 0$|⁠, we have |$\#\mathscr{G}_{\operatorname{CM}}(d) \ll_{\epsilon} d^{\epsilon}$| for all odd |$d$|⁠; on the other hand, for each |$A> 0$|⁠, we have |$\#\mathscr{G}_{\operatorname{CM}}(d) > (\log{d})^A$| for infinitely many odd |$d$|⁠.