  Let $\mathscr{G}_{\rm CM}(d)$ 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 $\mathscr{G}_{\rm CM}(d)$ 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 $\mathscr{G}_{\rm CM}(d') =
\mathscr{G}_{\rm CM}(d)$ possesses a well-defined, positive asymptotic density.
(2) Let $T_{\rm CM}(d) = \max_{G \in \mathscr{G}_{\rm CM}(d)} \#G$; under the
Generalized Riemann Hypothesis, $$\left(\frac{12e^{\gamma}}{\pi}\right)^{2/3}
\le \limsup_{\substack{d\to\infty\\d\text{ odd}}} \frac{T_{\rm
CM}(d)}{(d\log\log{d})^{2/3}} \le
\left(\frac{24e^{\gamma}}{\pi}\right)^{2/3}.$$ (3) For each $\epsilon > 0$, we
have $\#\mathscr{G}_{\rm CM}(d) \ll_{\epsilon} d^{\epsilon}$ for all odd $d$;
on the other hand, for each $A> 0$, we have $\#\mathscr{G}_{\rm CM}(d) >
(\log{d})^A$ for infinitely many odd $d$.
