Title: Quadratic forms and a local-global principle.
(joint with Julia Hartmann and Daniel Krashen)
Abstract:
Patching methods, which have been used to prove results about etale fundamental
groups, can be used to obtain a local-global principle for the existence of
F-rational points on homogeneous spaces, where F is the function field of a
curve over a discretely valued field. This in turn yields local-global
principles for quadratic forms and for central simple algebras; for quadratic
forms, the precise statements of these local-global principles depend on the
fundamental group of the graph associated to the closed fiber of a model of the
curve. Using these local-global principles, results can be obtained over
function fields regarding the u-invariant (concerning dimensions of anisotropic
quadratic forms) and the period-index problem for the Brauer group.