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 \'etale fundamental group $\pi$ of a curve $X$ defined over $K$. In the case that $K=\mathbb{Q}(\zeta_n)$ is the cyclotomic field and $X$ is the Fermat curve of degree $n \geq 3$, Anderson determined the action of $G_K$ on the \'etale homology with coefficients in $\mathbb{Z}/n \mathbb{Z}$.The \'etale homology is the first quotient in the lower central series of the \'etale fundamental group.In this paper, we determine the structure of the graded Lie algebra for $\pi$. 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 \'etale fundamental group of the Fermat curve of degree $n$, with coefficients in $\mathbb{Z}/n \mathbb{Z}$.