Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Local-global questions for tori over $p$-adic function fields

Authors: David Harari and Tamás Szamuely
Journal: J. Algebraic Geom. 25 (2016), 571-605
Published electronically: March 31, 2016
MathSciNet review: 3493592
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Abstract: We study local-global questions for Galois cohomology over the function field of a curve defined over a $p$-adic field (a field of cohomological dimension 3). We define Tate-Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of the base field coming from a closed point of the curve. In the case of a torus we establish a perfect duality between the first Tate-Shafarevich group of the torus and the second Tate-Shafarevich group of the dual torus. Building upon the duality theorem, we show that the failure of the local-global principle for rational points on principal homogeneous spaces under tori is controlled by a certain subquotient of a third étale cohomology group. We also prove a generalization to principal homogeneous spaces of certain reductive group schemes in the case when the base curve has good reduction.

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David Harari
Affiliation: Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France

Tamás Szamuely
Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda utca 13–15, H-1053 Budapest, Hungary – and – Central European University, Nádor utca 9, H-1051 Budapest, Hungary

Received by editor(s): October 4, 2013
Received by editor(s) in revised form: February 26, 2014
Published electronically: March 31, 2016
Article copyright: © Copyright 2016 University Press, Inc.