Arithmetic Orr invariants of absolute Galois groups

TL;DR

This paper introduces an arithmetic analogue of Orr invariants for Galois groups, connecting Galois actions on étale fundamental groups with topological invariants, and explores their relation to Grothendieck's section conjecture.

Contribution

It develops an arithmetic pro- extit{ extlangle}l extgreater{} Orr invariant and space, extending topological concepts to Galois groups, and analyzes their properties and relations to obstructions in section conjecture.

Findings

Determined the rank of the pro- extit{ extlangle}l extgreater{} Orr space as a -module.

Established a relation between the Orr invariants and Ellenberg's obstruction.

Extended Orr invariants to the setting of absolute Galois groups.

Abstract

Based on the analogies between mapping class groups and absolute Galois groups, we introduce an arithmetic pro-$\ell$ analogue of Orr invariants for a Galois element associated with Galois action on \'etale fundamental groups of punctured projective lines. At the same time, we also introduce pro-$\ell$ Orr space as an arithmetic analogue of Orr space whose third homotopy group is a target group of Orr invariant. We then determine its rank as $\mathbb{Z}_{\ell}$-module following Igusa-Orr's computation. Moreover, we investigate its relation with Ellenberg's obstruction to $\pi_1$-sections associated with lower central series filtration in the context of Grothendieck's section conjecture.