Existence of rational points as a homotopy limit problem

TL;DR

This paper connects the existence of rational points on smooth varieties over fields to homotopy fixed points of etale topological types, proposing a new model and linking to Grothendieck's Section Conjecture.

Contribution

It introduces a novel approach using homotopy fixed points to detect rational points and proposes a new model for the continuous etale homotopy fixed point space.

Findings

Existence of rational points relates to homotopy fixed points under Galois action.

A new model for the continuous etale homotopy fixed point space is defined.

Surjectivity in Grothendieck's Section Conjecture is linked to fixed points and homotopy fixed points.

Abstract

We show that the existence of rational points on smooth varieties over a field can be detected using homotopy fixed points of etale topological types under the Galois action. As our main example we show that the surjectivity statement in Grothendieck's Section Conjecture would follow from the surjectivity of the map from fixed points to continuous homotopy fixed points on the level of connected components. Along the way we define a new model for the continuous etale homotopy fixed point space of a smooth variety over a field under the Galois action.