A note on rank 1 log extendable isocrystals on simply connected open varieties

Abstract

In 2010 de Jong proposed a $p$-adic version of Gieseker's conjecture: if $X$ is a smooth, simply connected projective variety, then any isocrystal on $X$ is constant. This was proven by Esnault and Shiho under some additional assumptions. We show that the conjecture holds in the case of a non-proper variety with trivial tame fundamental group and for rank 1 log extendable isocrystals.