I will be talking about joint work with Mirela Ciperiani in which we describe methods for computing the division algebras which become split when the ground field is extended to the function field of a genus 1 curve. In particular, we show how these division algebras arise as cup products coming from the Weil pairing, generalizing previous computations by Haile, Han, and Wadsworth who computed these algebras for certain plane curves. If time permits, I will also describe some related work in progress concerning index reduction formulas for genus 1 curves (joint with Max Lieblich), which relies on a twisted (gerby) version of the Fourier-Mukai transform.