Torsion subgroups of CM elliptic curves over odd degree number fields

TL;DR

This paper classifies torsion subgroups of CM elliptic curves over odd degree number fields and explores their statistical properties, including density and growth bounds, under certain hypotheses.

Contribution

It provides a complete determination of torsion subgroup sets for odd degree fields and establishes new statistical theorems about their distribution and size.

Findings

The set of torsion groups for a fixed degree has positive asymptotic density.

Under GRH, the maximum torsion size grows roughly as (d log log d)^{2/3}.

The number of possible torsion groups is very small relative to d, but can be large infinitely often.

Abstract

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 $d$.