Remarks on cycle classes of sections of the arithmetic fundamental group

TL;DR

This paper introduces a new cycle class associated with sections of the arithmetic fundamental group of certain algebraic varieties, explores its algebraicity, and provides a novel proof of a key theorem relating to the index of curves over p-adic fields.

Contribution

It defines a cycle class for sections of the arithmetic fundamental group, investigates its algebraicity, and offers a new proof of Stix's theorem on the index of curves over p-adic fields.

Findings

The cycle class can be associated with sections of the fundamental group.

The class's algebraicity is established for curves over p-adic fields.

A new proof of Stix's theorem on the index of curves is provided.

Abstract

Given a smooth and separated K(pi,1) variety X over a field k, we associate a "cycle class" in etale cohomology with compact supports to any continuous section of the natural map from the arithmetic fundamental group of X to the absolute Galois group of k. We discuss the algebraicity of this class in the case of curves over p-adic fields, and deduce in particular a new proof of Stix's theorem according to which the index of a curve X over a p-adic field k must be a power of p as soon as the natural map from the arithmetic fundamental group of X to the absolute Galois group of k admits a section. Finally, an etale adaptation of Beilinson's geometrization of the pronilpotent completion of the topological fundamental group allows us to lift this cycle class in suitable cohomology groups.