Anatomy of torsion in the CM case

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 $$\mathbb {Q}$$ .