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

Paper

Abstract

For

a complete algebraically closed field, we construct a collection of non-isomorphic rank two

-local systems on

indexed by

. This implies that the de Jong fundamental group

depends on

and, if

has cardinality

, that

is not topologically countably generated. The argument in fact applies to any connected rigid analytic variety over

with a non-constant function to