Torsion points and Galois representations on CM elliptic curves

TL;DR

This paper investigates torsion points and Galois representations of CM elliptic curves over number fields, providing new bounds, classifications, and refinements of existing results in the context of complex multiplication.

Contribution

It offers refined degree calculations for rational torsion points, bounds on torsion subgroup sizes for nonmaximal orders, and a complete classification of torsion subgroups and cyclic isogenies for CM elliptic curves.

Findings

Computed degrees for rational N-torsion points, refining Silverberg's results

Bound the size of torsion subgroups for nonmaximal orders

Classified possible torsion subgroups and rational cyclic isogenies

Abstract

We prove several results on torsion points and Galois representations for complex multiplication (CM) elliptic curves over a number field containing the CM field. One result computes the degree in which such an elliptic curve has a rational point of order $N$, refining results of Silverberg. Another result bounds the size of the torsion subgroup of an elliptic curve with CM by a nonmaximal order in terms of the torsion subgroup of an elliptic curve with CM by the maximal order. Our techniques also yield a complete classification of both the possible torsion subgroups and the rational cyclic isogenies of a $K$-CM elliptic curve $E$ defined over $K(j(E))$.