The truth about torsion in the CM case, II

Abstract

Let $T_{\mathrm{CM}}(d)$ be the largest size of the torsion subgroup of an elliptic curve with complex multiplication (CM) defined over a degree $d$ number field. Work of Breuer and Clark--Pollack showed ${\mathrm{limsup}}_{d\to \infty }\frac{T_{\mathrm{CM}}(d)}{d\log \log d}\in (0,\infty )$. Here we show that the above limit supremum is precisely $\frac{e^3\ddot{I}€}{\sqrt{3}}$. We also study -- in part, out of necessity -- the upper order of the size of the torsion subgroup of various restricted classes of CM elliptic curves over number fields.

17 pages