Approximation forte pour les espaces homog\`enes de groupes semisimples sur le corps des fonctions d'une courbe alg\'ebrique complexe

TL;DR

This paper investigates the approximation properties of homogeneous spaces of semisimple algebraic groups over function fields of complex algebraic curves, establishing strong approximation results and identifying cases where it fails.

Contribution

It proves strong approximation for homogeneous spaces of semisimple groups over function fields, and demonstrates its failure for tori, advancing understanding of approximation in this context.

Findings

Strong approximation holds for homogeneous spaces of semisimple groups outside finite sets of places.

Strong approximation fails for tori over the same function fields.

Provides a clear distinction between the approximation behaviors of different algebraic groups.

Abstract

Let K be the function field of a curve over the complex field. Let X be a homogeneous space of a semisimple linear algebraic group. Strong approximation holds for X outside any finite nonempty set of places of K. Strong approximation fails for tori over K.