Raccord sur les espaces de Berkovich

TL;DR

The paper provides new geometric proofs that every finite group can be realized as a Galois group over certain fields associated with Berkovich spaces, extending classical results with more elementary methods.

Contribution

It introduces a geometric approach to Galois realizations over Berkovich spaces, offering simpler proofs of known theorems by Harbater.

Findings

Every finite group is a Galois group over $ ext{Ms}(B)(T)$.

New geometric proof of Galois realizations over $K(T)$ for valued fields.

Alternative proof of Harbater's theorem on Galois groups over convergent power series fields.

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 ${\bf Z}$.