University of Cambridge > Talks.cam > Isaac Newton Institute Seminar Series > Patching and a local-global principle for curves (joint with Julia Hartmann and Daniel Krashen)

Patching and a local-global principle for curves (joint with Julia Hartmann and Daniel Krashen)

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Non-Abelian Fundamental Groups in Arithmetic Geometry

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.

This talk is part of the Isaac Newton Institute Seminar Series series.

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