# Galois descent of semi-affinoid spaces

## TL;DR

This paper investigates the Galois descent of semi-affinoid non-archimedean analytic spaces, providing a formal model description, studying forms of analytic annuli, and applying results to resolutions of surface singularities in characteristic zero.

## Contribution

It introduces a method to describe formal models of semi-affinoid spaces via Galois fixed loci and Weil restrictions, and applies this to analyze annuli and surface singularities.

## Key findings

- Formal models of semi-affinoid spaces can be described using Galois fixed loci.
- A Weierstrass preparation theorem for annuli functions is established.
- Non-archimedean analytic proof of surface singularity resolutions is provided.

## Abstract

We study the Galois descent of semi-affinoid non-archimedean analytic spaces. These are the non-archimedean analytic spaces which admit an affine special formal scheme as model over a complete discrete valuation ring, such as for example open or closed polydiscs or polyannuli. Using Weil restrictions and Galois fixed loci for semi-affinoid spaces and their formal models, we describe a formal model of a $K$-analytic space $X$, provided that $X\otimes_KL$ is semi-affinoid for some finite tamely ramified extension $L$ of $K$. As an application, we study the forms of analytic annuli that are trivialized by a wide class of Galois extensions that includes totally tamely ramified extensions. In order to do so, we first establish a Weierstrass preparation result for analytic functions on annuli, and use it to linearize finite order automorphisms of annuli. Finally, we explain how from these results one can deduce a non-archimedean analytic proof of the existence of resolutions of singularities of surfaces in characteristic zero.