Dualit\'e et principe local-global sur des corps locaux de dimension 2

TL;DR

This paper develops duality theorems for Tate-Shafarevich groups over fields of Laurent series in two variables, and applies these to analyze local-global principles and weak approximation for algebraic groups over such fields.

Contribution

It introduces duality results for Tate-Shafarevich groups over two-variable Laurent series fields and applies them to local-global principles and weak approximation problems.

Findings

Duality theorems for Tate-Shafarevich groups of finite groups and tori.

Obstructions to local-global principles for torsors under linear algebraic groups.

Results on weak approximation for tori over $K_0$.

Abstract

Let $k$ be an algebraically closed field, a finite field or a $p$-adic field. Let $K_0=k((x,y))$ be the field of Laurent series in two variables over $k$. We define Tate-Shafarevich groups of a commutative group scheme over $K_0$ via cohomology classes locally trivial at each completion of $K_0$ coming from a codimension 1 point of $\text{Spec}\; k[[x,y]]$. We establish duality theorems between Tate-Shafarevich groups for finite groups schemes and for tori. We apply these results to the study of the obstruction to the local-global principle for $K_0$-torsors under a connected linear algebraic group, answering in that way a question of Colliot-Th\'el\`ene, Parimala and Suresh, and to the weak approximation for tori over $K_0$. Soit $k$ un corps alg\'ebriquement clos, un corps fini, ou encore un corps $p$-adique. Soit $K_0=k((x,y))$ le corps des s\'eries de Laurent \`a deux variables sur $k$. On d\'efinit les groupes de Tate-Shafarevich d'un $K_0$-sch\'ema en groupes commutatif en consid\'erant les classes de cohomologie qui deviennent triviales sur chaque compl\'et\'e de $K_0$ provenant d'un point codimension 1 de $\text{Spec}\; k[[x,y]]$. On \'etablit des th\'eor\`emes de dualit\'e arithm\'etique entre des groupes de Tate-Shafarevich pour les modules finis et pour les tores. On applique ces r\'esultats \`a l'\'etude du principe local-global pour les $K_0$-torseurs sous un groupe lin\'eaire connexe, r\'epondant ainsi \`a une question de Colliot-Th\'el\`ene, Parimala et Suresh, ainsi qu'\`a l'approximation faible pour les tores sur $K_0$.