Derived equivalences and isogenies of supersingular abelian varieties

We study (twisted) Fourier-Mukai partners of supersingular abelian varieties. We give a simple proof that any two supersingular abelian varieties of the same dimension at least two are derived isogenous, answering a question raised by Li, Lu, and Tang and generalizing the result of Li and Zou. Moreover, in every dimension greater than two, we construct derived-equivalent supersingular abelian varieties whose corresponding positive powers are never isomorphic. We also give conditions under which derived equivalence between two supersingular principally polarized abelian varieties implies that their underlying abelian varieties are isomorphic. In particular, this implication holds in dimensions at most three.