On fields of definition of torsion points of elliptic curves with complex multiplication

TL;DR

This paper investigates the Galois representations of elliptic curves with complex multiplication over rationals, providing classifications of torsion points' fields of definition and their relation to the CM field.

Contribution

It offers a detailed description of the mod p Galois representations for CM elliptic curves over rationals and classifies torsion points over Galois fields not containing the CM field.

Findings

Classification of Galois images for all primes p

Identification of cases with torsion points over fields not containing the CM field

Explicit description of fields of definition of torsion points

Abstract

For any elliptic curve E defined over the rationals with complex multiplication and for every prime p, we describe the image of the mod p Galois representation attached to E. We deduce information about the field of definition of torsion points of these curves, in particular we classify all cases where there are torsion points over Galois number fields not containing the field of definition of the CM.