Local-global principles for torsors over arithmetic curves
- 1 December 2015
- journal article
- research article
- Published by Project MUSE in American Journal of Mathematics
- Vol. 137 (6), 1559-1612
- https://doi.org/10.1353/ajm.2015.0039
Abstract
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.Keywords
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