Local-global principles for torsors over arithmetic curves
  We consider local-global principles for torsors under linear algebraic groups, over function fields of curves over complete discretely valued fields. The obstruction to such a principle is a version of the Tate-Shafarevich group; and for groups with rational components, we compute it explicitly and show that it is finite. This yields necessary and sufficient conditions for local-global principles to hold. Our results rely on first obtaining a Mayer-Vietoris sequence for Galois cohomology and then showing that torsors can be patched. We also give new applications to quadratic forms and central simple algebras. 
David Harbater, Julia Hartmann, and Daniel Krashen
math.NT math.AG math.RA