Torsion subgroups of CM elliptic curves over odd degree number fields

Let ${\mathcal{G}}_{\mathrm{C}\mathrm{M}}(d)$ 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 ${\mathcal{G}}_{\mathrm{C}\mathrm{M}}(d)$ 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 $d^{\mathrm{\prime }}$ with $\mathscr{G}{\rm CM}(d’) = \mathscr{G}{\rm CM}(d)$ possesses a well-defined, positive asymptotic density. (2) Let $T_{\mathrm{C}\mathrm{M}}(d)={\max }_{G\in {\mathcal{G}}_{\mathrm{C}\mathrm{M}}(d)}\mathrm{\#}G$; under the Generalized Riemann Hypothesis, $${\left(\frac{12e^{\gamma }}{\pi }\right)}^{2\mathrm{/}3}\le \underset{\begin{array}{c}d\to \mathrm{\infty }\\ d\text{ odd}\end{array}}{\mathrm{lim\hspace{0.17em}sup}}\frac{T_{\mathrm{C}\mathrm{M}}(d)}{(d\log \log d{)}^{2\mathrm{/}3}}\le {\left(\frac{24e^{\gamma }}{\pi }\right)}^{2\mathrm{/}3}\mathrm{.}$$ (3) For each $ϵ>0$, we have $\mathrm{\#}{\mathcal{G}}_{\mathrm{C}\mathrm{M}}(d){\ll }_{ϵ}{d}^{ϵ}$ for all odd $d$; on the other hand, for each $A>0$, we have $#\mathscr{G}_{\rm CM}(d) > (\log{d})^A$forinfinitelymanyodd$d$.