Abelian varieties isogenous to a power of an elliptic curve

Abstract

Let $E$ be an elliptic curve over a field $k$. Let $R:=\text{End}E$. There is a functor $\mathcal{H}{om}_R(-,E)$ from the category of finitely presented torsion-free left $R$-modules to the category of abelian varieties isogenous to a power of $E$, and a functor $\text{Hom}(-,E)$ in the opposite direction. We prove necessary and sufficient conditions on $E$ for these functors to be equivalences of categories.

21 pages, comments welcome

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