The 3rd MSJ-SI

The Mathematical Society of Japan, Seasonal Institute

Development of Galois-Teichmüller Theory
and Anabelian Geometry

October 28 Thursday, 16:00--16:50
D. Harbater (University of Pennsylvania)
Local-global principles over arithmetic curves
(Joint work with Julia Hartmann and Daniel Krashen.) The classical Tate-Shafarevich group Sha considers torsors for an abelian variety over a global field, and classifies those that become trivial at each completion. More generally, one may consider other fields F and other algebraic groups G (though Sha becomes just a pointed set if G is not commutative). This talk concerns the case in which G is a linear algebraic group that is rational (though possibly disconnected) over the function field F of a curve defined over a complete discretely valued field. In this situation, we show that Sha is finite, and we explicitly give its order in terms of the fundamental group of the reduction graph of a regular model of the curve and the maximal finite quotient of G. In particular, for such G, we show that a local-global principle holds if and only if either G is connected or the reduction graph is a tree. This has applications to the study of quadratic forms and central simple algebras.