When are two (gerbey) genus 1 curves equivalent?

Two algebraic varieties (or manifolds) are said to be derived equivalent if, roughly, they have the same set of functions on them. We discuss when two genus 1 curves (or genus 1 Riemann surfaces, if you prefer) are derived equivalent. This turns out to be related to when one of the curves is the classifying space of line bundles on the other curve. Background in moduli spaces and line bundles/Picard groups will not be assumed--I'll explain all the words in the abstract. Joint work with Soumya Sankar.