Lifts of supersingular abelian varieties with small Mumford-Tate groups

TL;DR

This paper explores the possibilities of lifting supersingular abelian varieties over finite fields to characteristic zero with small Mumford-Tate groups, revealing specific cases where such lifts are possible or impossible.

Contribution

It provides new results on lifting supersingular abelian varieties with small Mumford-Tate groups, including explicit constructions and obstructions.

Findings

Supersingular abelian surfaces can be lifted to isogenous squares of elliptic curves.

Supersingular abelian threefolds can be lifted to products of elliptic curves.

Supersingular abelian threefolds cannot be lifted to cubes of elliptic curves.

Abstract

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 threefolds, can be lifted to ones isogenous to a square, respectively product, of elliptic curves. On the other hand, we show that supersingular abelian threefolds cannot be lifted to one isogenous to the cube of an elliptic curve over the Witt vectors.