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

Abstract

Given an elliptic curve $E/k$ and a Galois extension $k^{\prime }/k$, we construct an exact functor from torsion-free modules over the endomorphism ring $\mathrm{End}(E_{k^{\prime }})$ with a semilinear $\mathrm{Gal}(k^{\prime }/k)$ action to abelian varieties over $k$ that are $k^{\prime }$-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.

6 pages, added references