The Galois action on the lower central series of the fundamental group of the Fermat curve

Abstract

Information about the absolute Galois group $G_K$ of a number field $K$ is encoded in how it acts on the étale fundamental group $\ddot{I}€$ of a curve $X$ defined over $K$. In the case that $K=\mathbb{Q}(\hat{I}{\P }_n)$ is the cyclotomic field and $X$ is the Fermat curve of degree $n\ge 3$, Anderson determined the action of $G_K$ on the étale homology with coefficients in $\mathbb{Z}/n\mathbb{Z}$.The étale homology is the first quotient in the lower central series of the étale fundamental group.In this paper, we determine the structure of the graded Lie algebra for $\ddot{I}€$. As a consequence, this determines the action of $G_K$ on all degrees of the associated graded quotient of the lower central series of the étale fundamental group of the Fermat curve of degree $n$, with coefficients in $\mathbb{Z}/n\mathbb{Z}$.