Trading degree for dimension in the section conjecture: The non-abelian Shapiro Lemma

TL;DR

This paper investigates the section conjecture in anabelian geometry, demonstrating how Weil restriction affects the etale fundamental group through a non-abelian Shapiro Lemma, thus providing new evidence for the conjecture.

Contribution

It introduces a non-abelian Shapiro Lemma for Weil restrictions, linking the etale fundamental groups in a novel way that supports the section conjecture.

Findings

Determines the etale fundamental group of Weil restrictions.

Establishes a non-abelian Shapiro Lemma for group cohomology.

Provides evidence supporting the section conjecture.

Abstract

This note aims at providing evidence for the section conjecture of anabelian geometry by establishing its behaviour under Weil restriction of scalars. In particular, the etale fundamental group of the Weil restriction is determined by means of a Shapiro Lemma for non-abelian group cohomology.