Existence of rational points as a homotopy limit problem

2013/09/02 by Quick, Gereon · 1 citation
#14F35 #14G05 #55P91 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.1309.0463

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.

Cited by

• On the real Section Conjecture in étale homotopy theory

Related

• Homotopy Obstructions to Rational Points
• Vari'et'es presque rationnelles, leurs points rationnels et leurs d'eg'en'erescences
• Brauer and Etale Homotopy Obstructions to Rational Points on Open Covers
• Homotopy Rational Points of Brauer-Severi Varieties
• Étale Homotopy Obstructions of Arithmetic Spheres