L'espace ad\'elique d'un tore sur un corps de fonctions

TL;DR

This paper investigates the adelic points space of a torus over a function field of a curve, analyzing its structure and quotient in various base fields, and establishing the discreteness of rational points within this space.

Contribution

It provides a detailed description of the adelic points space and its quotient for tori over function fields, extending understanding in different base field cases.

Findings

The rational points form a discrete subgroup in the adelic points space.

Explicit descriptions of the quotient space are given for algebraically closed, Laurent series, and p-adic base fields.

The structure of adelic points varies significantly depending on the base field.

Abstract

Let $k$ be a field of characteristic 0 and let $K$ be the function field of a smooth projective geometrically integral $k$-curve $X$. Let $T$ be a $K$-torus. In this article, we aim at studying the space of adelic points $T(S,\mathbb{A}_K)$ of $T$ outside a finite set $S$ of closed points of $X$. We start by proving that the group $T(K)$ of rational points of $T$ is always discrete (hence closed) in $T(S,\mathbb{A}_K)$. We then describe the quotient $T(\emptyset,\mathbb{A}_K)/T(K)$ in each of the following three cases: $k$ is an algebraically closed field, $k$ is the field of Laurent series $\mathbb{C}((t))$, and $k$ is a $p$-adic field. Soient $k$ un corps de caract\'eristique 0 et $K$ le corps des fonctions d'une $k$-courbe projective lisse g\'eom\'etriquement int\`egre $X$. Soit $T$ un $K$-tore. Dans cet article, on cherche \`a \'etudier l'espace des points ad\'eliques $T(S,\mathbb{A}_K)$ de $T$ hors d'un ensemble fini $S$ de points ferm\'es de $X$. On commence par montrer que le groupe $T(K)$ des points rationnels de $T$ est toujours ferm\'e discret dans $T(S,\mathbb{A}_K)$. On d\'ecrit ensuite le quotient $T(\emptyset,\mathbb{A}_K)/T(K)$ dans chacun des trois cas suivants: $k$ corps alg\'ebriquement clos, $k=\mathbb{C}((t))$ et $k$ corps $p$-adique.