REAL

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

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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 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.

Item Type: Article
Uncontrolled Keywords: Kernel; 1-MOTIVES; COMPLEXES; cohomology; CURVES; DUALITY THEOREMS; HASSE PRINCIPLE; BLOCH-KATO CONJECTURE; LINEAR ALGEBRAIC-GROUPS
Subjects: Q Science / természettudomány > QA Mathematics / matematika
Q Science / természettudomány > QA Mathematics / matematika > QA72 Algebra / algebra
SWORD Depositor: MTMT SWORD
Depositing User: MTMT SWORD
Date Deposited: 03 Jan 2017 14:01
Last Modified: 03 Jan 2017 14:02
URI: http://real.mtak.hu/id/eprint/44233

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