We study local-global principles for torsors under reductive linear algebraic groups over semi-global fields; that is, over one-variable function fields over complete discretely valued fields. We provide conditions on the group and the semi-global field under which the local-global principle holds, and we compute the obstruction to the local-global principle in certain classes of examples. Using our description of the obstruction, we give the first example of a semisimple simply connected group over a semi-global field where the local-global principle fails. Our methods include patching and R-equivalence.
To Gopal Prasad on his 75th birthday
"Local-Global Principles for Constant Reductive Groups over Semi-Global Fields." Michigan Math. J. 72 77 - 144, August 2022. https://doi.org/10.1307/mmj/20217219