Galois action on homology of generalized Fermat Curves

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.