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

Harari, D. and Szamuely, Tamás (2016) Local-global questions for tori over p-adic function fields. JOURNAL OF ALGEBRAIC GEOMETRY, 25 (3). pp. 571-605. ISSN 1056-3911

Preview

Text 1307.4782.pdf Download (351kB) | Preview

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 etale 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.

Actions (login required)

Edit Item