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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
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