Galois Action on Homology of the Heisenberg Curve

TL;DR

This paper studies the topological and algebraic properties of the Heisenberg curve, focusing on its fundamental group, homology, and the action of Artin's Braid group, revealing new insights into its structure.

Contribution

It computes the fundamental group of the Heisenberg curve and describes its homology via irreducible representations, linking topology and algebra in a novel way.

Findings

Fundamental group of the Heisenberg curve is explicitly computed.

Homology is described using irreducible representations of the Heisenberg group.

Artin's Braid group acts naturally on the curve and its homology.

Abstract

The Heisenberg curve is defined topologically as a cover of the Fermat curve and corresponds to an extension of the projective line minus three points by the non-abelian Heisenberg group modulo n. We compute its fundamental group and investigate an action from Artin's Braid group to the curve itself and its homology. We also provide a description of the homology in terms of irreducible representations of the Heisenberg group over a field of characteristic $0$.