The truth about torsion in the CM case

TL;DR

This paper establishes an upper bound on the size of the torsion subgroup of CM elliptic curves over degree d number fields, showing it grows roughly as d log log d.

Contribution

It provides a precise asymptotic upper bound for torsion subgroups in CM elliptic curves over number fields, clarifying their growth rate.

Findings

Torsion subgroup size is bounded by d log log d for CM elliptic curves.

The result improves understanding of torsion growth in CM elliptic curves.

Provides a key estimate for future research in elliptic curve torsion properties.

Abstract

We show that the upper order of the size of the torsion subgroup of a CM elliptic curve over a degree d number field is d log log d.