The de Jong fundamental group of $\mathbb{P}^1_C$ depends on $C$ and is not always topologically countably generated

TL;DR

This paper demonstrates that the de Jong fundamental group of the projective line over certain algebraically closed fields varies with the field and can lack topological countable generation, extending to broader rigid analytic varieties.

Contribution

It establishes the dependence of the de Jong fundamental group on the base field and shows it may not be topologically countably generated in large cardinality cases, generalizing previous results.

Findings

The de Jong fundamental group depends on the base field C.

For large C, the group is not topologically countably generated.

Results extend to connected rigid analytic varieties with non-constant maps.

Abstract

For $C/\mathbb{Q}_p$ complete and algebraically closed, we show that the de Jong fundamental group $\pi_{1,\mathrm{dJ}}(\mathbb{P}^1_C)$ depends on $C$ and, if $C$ has cardinality $>2^{\mathbb{N}}$, that it is not topologically countably generated. The result and proofs generalize to any connected rigid analytic variety with a non-constant map to $\mathbb{P}^n_C$.