Abelian varieties isogenous to a power of an elliptic curve

TL;DR

This paper characterizes when certain functors establish an equivalence between modules over the endomorphism ring of an elliptic curve and abelian varieties isogenous to powers of that curve.

Contribution

It provides necessary and sufficient conditions on an elliptic curve for the functors to be categorical equivalences.

Findings

Identifies conditions for functor equivalence

Connects module categories with abelian varieties

Advances understanding of elliptic curve endomorphisms

Abstract

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.