The truth about torsion in the CM case, II

TL;DR

This paper precisely determines the asymptotic growth rate of the largest torsion subgroup size of CM elliptic curves over number fields, refining previous bounds and providing exact limit supremum.

Contribution

It establishes the exact value of the limit supremum for the torsion subgroup size of CM elliptic curves over number fields, improving upon earlier bounds.

Findings

The limit supremum of torsion subgroup size ratio is exactly .567...

Provides detailed analysis of torsion sizes in restricted classes of CM elliptic curves

Refines understanding of torsion growth in CM elliptic curves over number fields.

Abstract

Let $T_{{\rm CM}}(d)$ 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 $\limsup_{d \to \infty} \frac{T_{{\rm CM}}(d)}{d \log \log d} \in (0,\infty)$. Here we show that the above limit supremum is precisely $\frac{e^{\gamma} \pi}{\sqrt{3}}$. 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.