and Anabelian Geometry

October 28 Thursday, 16:00--16:50 | |

D. Harbater
(University of Pennsylvania)
Local-global principles over arithmetic curves
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(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. |