## Torsion subgroups of CM elliptic curves over odd degree number fields.(English)Zbl 1405.11072

Summary: Let $$\mathcal G_{\mathrm{CM}}(d)$$ 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 $$\mathcal G_{\mathrm{CM}}(d)$$ 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 $$d'$$ with $$\mathcal G_{\mathrm{CM}}(d')=\mathcal G_{\mathrm{CM}}(d)$$ 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, $\left(\frac{12 \mathrm{e}^\gamma}{\pi}\right)^{2/3} \leq \limsup\limits_{\begin{pmatrix} d \rightarrow\infty \\ d\,\mathrm{odd} \end{pmatrix}} \frac{T_{\mathrm{CM}}(d)}{(d\log\log d)^{2/3}} \leq\left(\frac{24\mathrm{e}^\gamma}{\pi}\right)^{2/3}.$ (3) For each $$\in >0$$, we have $$\#\mathcal G_{\mathrm{CM}}(d) \ll_\in d^\epsilon$$ 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 $$d$$.

### MSC:

 11G05 Elliptic curves over global fields 11G15 Complex multiplication and moduli of abelian varieties

### Keywords:

complex multiplication; elliptic curves; torsion subgroups
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