Abelian varieties isogenous to a power of an elliptic curve

Bruce W. Jordan, Allan G. Keeton, Bjorn Poonen, Eric M. Rains, Nicholas Shepherd-Barron, John T. Tate

Published 2016-02-19Version 1

Let $E$ be an elliptic curve over a field $k$. Let $R:= \text{End}\, E$. There is a functor $\mathscr{H}\!\!\mathit{om}_R(-,E)$ 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.