Raccord sur les espaces de Berkovich

Abstract

Let $X$ be a Berkovich space over a valued field. We prove that every finite group is a Galois group over $\Ms(B)(T)$, where $\Ms(B)$ is the field of meromorphic functions over a part $B$ of $X$ satisfying some conditions. This gives a new geometric proof that every finite group is a Galois group over $K(T)$, where $K$ is a complete valued field with non-trivial valuation. Then we switch to Berkovich spaces over ${\bf Z}$ and use a similar strategy to give a new proof of the following theorem by D. Harbater: every finite group is a Galois group over a field of convergent arithmetic power series. We believe our proof to be more geometric and elementary that the original one. We have included the necessary background on Berkovich spaces over ${\bf Z}$.