Derived categories of torsors for abelian schemes

In the first part of our paper, we show that there exist non-isomorphic derived equivalent genus $1$ curves, and correspondingly there exist non-isomorphic moduli spaces of stable vector bundles on genus $1$ curves in general. Neither occurs over an algebraically closed field. We give necessary and sufficient conditions for two genus $1$ curves to be derived equivalent, and we go on to study when two principal homogeneous spaces for an abelian variety have equivalent derived categories. We apply our results to study twisted derived equivalences of the form $D^b(J,\alpha)\simeq D^b(J,\beta)$, when $J$ is an elliptic fibration, giving a partial answer to a question of C\u{a}ld\u{a}raru.