Th\'eor\`emes de dualit\'e pour les corps de fonctions sur des corps locaux sup\'erieurs et applications arithm\'etiques

TL;DR

This paper develops duality theorems for Tate-Shafarevich groups over function fields of curves on higher local fields, with applications to local-global principles and obstructions in arithmetic geometry.

Contribution

It establishes new duality theorems for Tate-Shafarevich groups of various algebraic groups over these function fields, extending previous results and applying them to arithmetic problems.

Findings

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

Results on weak approximation for tori over the function field.

Counterexamples to the local-global principle for central simple algebras.

Abstract

Let $K$ be the function field of a smooth projective curve $X$ over a higher-dimensional local field $k$. We define Tate-Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of $K$ coming from a closed point of $X$. We establish duality theorems between Tate-Shafarevich groups for finite groups schemes, for tori, for groups of multiplicative type, and even for 2-term complexes of tori. We apply these results to the weak approximation for tori over $K$ and to the study of the obstruction to the local-global principle for $K$-torsors under a connected linear algebraic group. We also give examples and counter-examples to the local-global principle for central simple algebras over $K$. Soit $K$ le corps des fonctions d'une courbe projective lisse X sur un corps local sup\'erieur $k$. On d\'efinit les groupes de Tate-Shafarevich d'un sch\'ema en groupes commutatif en consid\'erant les classes de cohomologie qui deviennent triviales sur chaque compl\'et\'e de $K$ provenant d'un point ferm\'e de $X$. On \'etablit des th\'eor\`emes de dualit\'e arithm\'etique entre des groupes de Tate-Shafarevich pour les modules finis, pour les tores, pour les groupes de type multiplicatif, et m\^eme pour les complexes \`a deux termes de tores. On applique ces r\'esultats \`a l'approximation faible pour les tores sur $K$ et \`a l'\'etude du principe local-global pour les $K$-torseurs sous un groupe lin\'eaire connexe. On exhibe aussi des exemples et des contre-exemples au principe local-global pour les alg\`ebres simples centrales sur $K$.