The \'etale topos reconstructs varieties over sub-p-adic fields

Abstract

Let $K$ be a sub-$p$-adic field. We show that the functor sending a finite type $K$-scheme to its \'etale topos is fully faithful after localizing at the class of universal homeomorphisms. This generalizes a result of Voevodsky, who proved the analogous theorem for fields finitely generated over $\mathbb{Q}$. Our proof relies on Mochizuki's Hom-theorem in anabelian geometry, and a study of point-theoretic morphisms of fundamental groups of curves.