Torsion for CM elliptic curves defined over number fields of degree 2p

TL;DR

This paper classifies possible torsion subgroups of CM elliptic curves over degree 2p number fields, linking the problem to the existence of infinitely many Sophie Germain primes.

Contribution

It provides a characterization of torsion groups for CM elliptic curves over degree 2p fields, connecting the classification to prime number conjectures.

Findings

Identifies which torsion groups can occur for CM elliptic curves over these fields.

Shows the classification depends on the existence of infinitely many Sophie Germain primes.

Highlights the deep connection between elliptic curve torsion structures and prime number theory.

Abstract

For a prime number p, we characterize the groups that may arise as torsion subgroups of an elliptic curve with complex multiplication defined over a number field of degree 2p. In particular, our work shows that a classification in the strongest sense is tied to determining whether there exist infinitely many Sophie Germain primes.