Abelian varieties isogenous to a power of an elliptic curve

Abstract

Let be an elliptic curve over a field k. Let [mathematical notation] . There is a functor [mathematical notation] 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 Hom (-, E) in the opposite direction. We prove necessary and sufficient conditions on for these functors to be equivalences of categories. We also prove a partial generalization in which E is replaced by a suitable higher-dimensional abelian variety over F[subscript P].