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

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 étale 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).

22 pages