Torsion subgroups of CM elliptic curves over odd degree number fields

Abstract

Let ${\mathcal{G}}_{\mathrm{CM}}(d)$ denote the collection of groups (up to isomorphism) that appear as the torsion subgroup of a CM elliptic curve over a degree $d$ number field. We completely determine ${\mathcal{G}}_{\mathrm{CM}}(d)$ 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 $d^{\prime }$ with ${\mathcal{G}}_{\mathrm{CM}}(d^{\prime })={\mathcal{G}}_{\mathrm{CM}}(d)$ possesses a well-defined, positive asymptotic density. (2) Let $T_{\mathrm{CM}}(d)={\max }_{G\in {\mathcal{G}}_{\mathrm{CM}}(d)}\#G$; under the Generalized Riemann Hypothesis, $${\left(\frac{12e^3}{\ddot{I}}€\right)}^{2/3}\le \underset{\begin{array}{c}d\to \infty \\ d\text{ odd}\end{array}}{\mathrm{limsup}}\frac{T_{\mathrm{CM}}(d)}{(d\log \log d{)}^{2/3}}\le {\left(\frac{24e^3}{\ddot{I}}€\right)}^{2/3}.$$ (3) For each $\hat{I}\mu >0$, we have $\#{\mathcal{G}}_{\mathrm{CM}}(d){\ll }_{\hat{I}}\mu d^{\hat{I}}\mu$ for all odd $d$; on the other hand, for each $A>0$, we have $\#{\mathcal{G}}_{\mathrm{CM}}(d)>(\log d{)}^A$ for infinitely many odd $d$.

26 pages