$\text{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$ as a geometric fundamental group

TL;DR

This paper proves that the absolute Galois group of p-adic numbers can be realized as the étale fundamental group of a geometric object, linking local Galois representations to local systems on a perfectoid space.

Contribution

It establishes a geometric interpretation of the Galois group of $ ext{Q}_p$ as an étale fundamental group of a specific algebraic object, combining p-adic Hodge theory and tilting techniques.

Findings

Galois group of $ ext{Q}_p$ is the étale fundamental group of a certain algebraic object.

Local Galois representations correspond to local systems on this object.

Construction uses perfectoid spaces, the fundamental curve, and tilting equivalence.

Abstract

Let $p$ be a prime number. In this article we present a theorem, suggested by Peter Scholze, which states that the absolute Galois group of $\mathbf{Q}_p$ is the \'etale fundamental group of a certain object $Z$ which is defined over an algebraically closed field. Thus, local Galois representations correspond to local systems on $Z$. In brief, $Z$ is a (non-representable) quotient of a perfectoid space. The construction combines two themes: the fundamental curve of $p$-adic Hodge theory (due to Fargues-Fontaine) and the tilting equivalence (due to Scholze).