Galois action on homology of generalized Fermat Curves

TL;DR

This paper studies the Galois actions on the homology of generalized Fermat curves, revealing their structure and connections to Galois representations and the Burau representation.

Contribution

It computes the fundamental group of these curves and analyzes the Galois and Galois cover actions on their homology, unifying various aspects of their symmetry.

Findings

Computed the fundamental group of generalized Fermat curves.

Analyzed Galois and absolute Galois group actions on homology.

Explored connections to the pro-ℓ Burau representation.

Abstract

The fundamental group of Fermat and generalized Fermat curves is computed. These curves are Galois ramified covers of the projective line with abelian Galois groups $H$. We provide a unified study of the action of both cover Galois group $H$ and the absolute Galois group $\mathrm{Gal}(\bar{\Q}/\Q)$ on the pro-$\ell$ homology of the curves in study. Also the relation to the pro-$\ell$ Burau representation is investigated.