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

Abstract

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