Endomorphisms of power series fields and residue fields of Fargues-Fontaine curves

TL;DR

This paper investigates endomorphisms of power series fields and their impact on the structure of Fargues-Fontaine curves, revealing non-bijective endomorphisms and non-isomorphic residue fields, thus addressing open questions in p-adic geometry.

Contribution

It demonstrates the existence of non-bijective endomorphisms of completed algebraic closures of power series fields and resolves a question about residue fields of Fargues-Fontaine curves.

Findings

Existence of non-bijective endomorphisms of completed algebraic closures of k((t))

Existence of points on Fargues-Fontaine curves with residue fields not isomorphic to C_p

Resolution of a question by Fargues and Fontaine regarding residue fields

Abstract

We show that for k a perfect field of characteristic p, there exist endomorphisms of the completed algebraic closure of k((t)) which are not bijective. As a corollary, we resolve a question of Fargues and Fontaine by showing that for p a prime and C_p a completed algebraic closure of Q_p, there exist closed points of the Fargues-Fontaine curve associated to C_p whose residue fields are not (even abstractly) isomorphic to C_p as topological fields.