Simple connectivity of Fargues--Fontaine curves

TL;DR

This paper proves that Fargues--Fontaine curves over algebraically closed fields of characteristic p are geometrically simply connected, and extends this result to perfectoid spaces, impacting the understanding of their fundamental groups.

Contribution

It establishes the geometric simple connectivity of Fargues--Fontaine curves and generalizes Drinfeld's lemma to perfectoid spaces.

Findings

Fargues--Fontaine curves are geometrically simply connected.

No nontrivial finite étale covers exist after base extension.

Extension of Drinfeld's lemma to perfectoid spaces.

Abstract

We show that the Fargues--Fontaine curve associated to an algebraically closed field of characteristic p is geometrically simply connected; that is, its base extension from Q_p to any complete algebraically closed overfield admits no nontrivial connected finite etale covering. We then deduce from this an analogue for perfectoid spaces (and some related objects) of Drinfeld's lemma on the fundamental group of a product of schemes in characteristic p.