\'Etale homotopy equivalence of rational points on algebraic varieties

TL;DR

This paper explores the concept of étale homotopy equivalence of rational points on algebraic varieties, establishing conditions under which points are equivalent over different fields and comparing this to other equivalence relations.

Contribution

It introduces a relative version of the étale homotopy type to analyze rational points and characterizes their equivalence over p-adic, real, and number fields, revealing new relations.

Findings

Over p-adic fields, rational points are homotopy equivalent iff they are étale-Brauer equivalent.

Over the real field, rational points are étale homotopy equivalent iff in the same connected component.

Over number fields, this equivalence is finer than other known relations for certain Châtelet surfaces.

Abstract

It is possible to talk about the \'etale homotopy equivalence of rational points on algebraic varieties by using a relative version of the \'etale homotopy type. We show that over $p$-adic fields rational points are homotopy equivalent in this sense if and only if they are \'etale-Brauer equivalent. We also show that over the real field rational points on projective varieties are \'etale homotopy equivalent if and only if they are in the same connected component. We also study this equivalence relation over number fields and prove that in this case it is finer than the other two equivalence relations for certain generalised Ch\^atelet surfaces.