Patching and a local-global principle for curves (joint with Julia Hartmann and Daniel Krashen)
Seminar Room 1, Newton Institute
Using patching, we establish a local-global principle for actions of algebraic groups that are defined over the function field of a curve over a complete discretely valued field. This result has applications to quadratic forms and to Brauer groups.
In the case of quadratic forms, we obtain a result on the u-invariant of function fields, in particular reproving the theorem of Parimala and Suresh that the u-invariant of a one-variable p-adic function field is 8. Concerning Brauer groups, we obtain results on the period-index problem for such fields, in particular reproving a result of Lieblich. We also obtain local-global principles for quadratic forms and for Brauer groups.
Our local-global principle for group actions holds in general for connected rational groups. In the disconnected case, the validity of the principle depends on the topology of a graph associated to the closed fiber of a model of the curve. The fundamental group of this graph is isomorphic to a certain quotient of the etale fundamental group; and the local-global principle holds even for disconnected rational groups if and only if the graph is a tree. In that case, the local-global principle for quadratic forms can be strengthened. In general, the cohomology of the graph determines the kernel of the local-global map on Witt groups.