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be an elliptic curve over a field
. There is a functor
from the category of finitely presented torsion-free left
-modules to the category of abelian varieties isogenous to a power of
, and a functor
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
is replaced by a suitable higher-dimensional abelian variety over
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