  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.
