Mathematische Zeitschrift

, Volume 285, Issue 3, pp 795–820

Anatomy of torsion in the CM case

Article

DOI: 10.1007/s00209-016-1727-5

Cite this article as:
Bourdon, A., Clark, P.L. & Pollack, P. Math. Z. (2017) 285: 795. doi:10.1007/s00209-016-1727-5
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Abstract

Let \(T_{\mathrm {CM}}(d)\) denote the maximum size of a torsion subgroup of a CM elliptic curve over a degree d number field. We initiate a systematic study of the asymptotic behavior of \(T_{\mathrm {CM}}(d)\) as an arithmetic function. Whereas a recent result of the last two authors computes the upper order of \(T_{\mathrm {CM}}(d)\), here we determine the lower order, the typical order and the average order of \(T_{\mathrm {CM}}(d)\) as well as study the number of isomorphism classes of groups G of order \(T_{\mathrm {CM}}(d)\) which arise as the torsion subgroup of a CM elliptic curve over a degree d number field. To establish these analytic results we need to extend some prior algebraic results. Especially, if \(E_{/F}\) is a CM elliptic curve over a degree d number field, we show that d is divisible by a certain function of \(\# E(F)[{\text {tors}}]\), and we give a complete characterization of all degrees d such that every torsion subgroup of a CM elliptic curve defined over a degree d number field already occurs over \(\mathbb {Q}\).

Mathematics Subject Classification

Primary 11G15 Secondary 11G05 11N25 11N37 

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Mathematics Department, Boyd Graduate Studies Research CenterUniversity of GeorgiaAthensUSA

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