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

TL;DR

This paper proves that over sub-$p$-adic fields, the functor from finite type schemes to their étale topos is fully faithful after certain localizations, extending Voevodsky's result to a broader class of fields.

Contribution

It generalizes Voevodsky's theorem by establishing full faithfulness of the étale topos functor over sub-$p$-adic fields, using Mochizuki's Hom-theorem and fundamental group analysis.

Findings

The étale topos functor is fully faithful after localization at universal homeomorphisms.

Extension of Voevodsky's theorem to sub-$p$-adic fields.

Application of anabelian geometry techniques to scheme reconstruction.

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.