# Abelian varieties isogenous to a power of an elliptic curve over a Galois extension

## TL;DR

This paper constructs a functor linking modules with Galois actions to abelian varieties isogenous to powers of an elliptic curve, and applies it to show CM elliptic curves are isogenous over the ground field to those with maximal order CM.

## Contribution

It introduces a functor from modules with Galois actions to abelian varieties, providing new insights into the structure of CM elliptic curves over Galois extensions.

## Key findings

- Constructed an exact functor from modules with Galois action to abelian varieties.
- Proved that every CM elliptic curve over a field is isogenous to one with maximal order CM.
- Established a connection between endomorphism modules and isogeny classes of elliptic curves.

## Abstract

Given an elliptic curve $E/k$ and a Galois extension $k'/k$, we construct an exact functor from torsion-free modules over the endomorphism ring ${\rm End}(E_{k'})$ with a semilinear ${\rm Gal}(k'/k)$ action to abelian varieties over $k$ that are $k'$-isogenous to a power of $E$. As an application, we show that every elliptic curve with complex multiplication geometrically is isogenous over the ground field to one with complex multiplication by a maximal order.