Lifts of Supersingular Abelian Varieties With Small Mumford-Tate Groups

We investigate to what extent an abelian variety over a finite field can be lifted to one in characteristic zero with small Mumford-Tate group. We prove that supersingular abelian surfaces, respectively three-folds, can be lifted to ones isogenous to a square, respectively product, of elliptic curves. On the other hand, we show that supersingular abelian three-folds cannot be lifted to one isogenous to the cube of an elliptic curve over the Witt vectors.