Quelques remarques \`a propos d'un th\'eor\`eme de Checcoli

TL;DR

This paper explores the generalization of Checcoli's theorem on bounded local degrees from number fields to function fields of positive characteristic, highlighting the necessity of certain hypotheses through examples.

Contribution

It extends Checcoli's result to function fields under specific conditions and demonstrates the importance of the prime-to-p hypothesis with a counterexample.

Findings

An analogue of Checcoli's theorem in positive characteristic with the prime-to-p condition.

A counterexample showing the necessity of the prime-to-p hypothesis.

Clarification of limitations in generalizing number field results to function fields.

Abstract

In his thesis, S. Checcoli shows that, among other results, if $K$ is a number field and if $L/K$ is an infinite Galois extension with Galois group $G$ of finite exponent, then $L$ has uniformly bounded local degrees at every prime of $K$. In this article we gather two remarks about the generalisation of S. Checcoli's result to function fields of positive characteristic. We first show an analogue of her theorem $2.2.2$ in this context, under the hypothesis that the Galois group exponent is prime to $p$. Using an example, we then show that this hypothesis is in fact necesary.---Dans sa th\`ese, S. Checcoli montre, entre autres r\'esultats, que si K est un corps de nombres et si L=K est une extension galoisienne in finie de groupe de Galois G d'exposant fini, alors les degr\'es locaux sur L sont uniform\'ement born\'es en toutes les places de K. Dans cette article nous rassemblons deux remarques \`a propos de la g\'en\'eralisation du r\'esultat de S. Checcoli aux corps de fonctions de caract\'eristique positive. D'une part nous montrons un analogue de son th\'eor\`eme dans ce cadre, sous l'hypoth\`ese que l'exposant du groupe de Galois soit premier \`a p. D'autre part, nous montrons \`a l'aide d'un exemple que cette hypoth\`ese est en fait n\'ecessaire.