Explicit descent on elliptic curves and splitting Brauer classes

We prove new results on splitting Brauer classes by genus 1 curves, settling in particular the case of degree 7 classes over global fields. Though our method is cohomological in nature, and proceeds by considering the more difficult problem of splitting $\mu_N$-gerbes, we use crucial input from the arithmetic of modular curves and explicit $N$-descent on elliptic curves.