Mathematische Zeitschrift

, Volume 285, Issue 3–4, pp 795–820 | Cite as

Anatomy of torsion in the CM case

  • Abbey Bourdon
  • Pete L. Clark
  • Paul Pollack


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 



We thank Filip Najman for explaining how to obtain stronger lower bounds on T(6) and T(10). We thank Robert S. Rumely for suggesting we investigate prime power Olson degrees. The exposition in Sect. 7 benefitted from talks by Carl Pomerance on the material in [8]. The first author was supported in part by NSF Grant DMS-1344994 (RTG in Algebra, Algebraic Geometry, and Number Theory, at the University of Georgia). The third author is supported by NSF award DMS-1402268.


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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

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

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