Vari\'et\'es ab\'eliennes sur les corps de fonctions de courbes sur des corps locaux sup\'erieurs

TL;DR

This paper develops duality theorems for Tate-Shafarevich groups of abelian varieties over function fields of curves defined over higher-dimensional local fields, extending classical duality results.

Contribution

It introduces new duality theorems for Tate-Shafarevich groups over higher-dimensional local fields, generalizing existing theories to more complex base fields.

Findings

Established local duality theorems for abelian varieties over higher-dimensional local fields.

Proved global duality theorems for Tate-Shafarevich groups over function fields.

Extended classical duality results to higher-dimensional local field contexts.

Abstract

Let $k$ be a higher-dimensional local field and $X$ be a smooth projective geometrically integral curve over $k$. Let $K$ be the function field of $X$. We define Tate-Shafarevich groups of an abelian variety via cohomology classes locally trivial at each completion of $K$ coming from a closed point of $X$. We prove local duality theorems for abelian varieties over $k$, as well as global duality theorems for Tate-Shafarevich groups of abelian varieties over $K$. Soient $k$ un corps local sup\'erieur et $X$ une courbe projective lisse g\'eom\'etriquement int\`egre de corps de fonctions $K$. On d\'efinit les groupes de Tate-Shafarevich d'une vari\'et\'e ab\'elienne 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 locale pour les vari\'et\'es ab\'eliennes sur $k$, ainsi que des th\'eor\`emes de dualit\'e globale pour les groupes de Tate-Shafarevich des vari\'et\'es ab\'eliennes sur $K$.