Raccord sur les espaces de Berkovich

Details Raccord sur les espaces de Berkovich Raccord sur les espaces de Berkovich

2008-09-22

arxiv.org/abs/0809.3656v4

Algebra & Number Theory 4 (2010), n{\deg} 3, p.297-334

Mathematics

Number Theory

45 pages, 3 figures, in French; v4: final version. To be published in Algebra & Number Theory

Scientific paper

10.2140/ant.2010.4.297

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 ${\bf Z}$.

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