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

TL;DR

This paper investigates the Galois group action on the lower central series of the étale fundamental group of Fermat curves over cyclotomic fields, extending previous results on homology to higher degrees.

Contribution

It determines the structure of the graded Lie algebra associated with the fundamental group, revealing the Galois action on all degrees of the lower central series.

Findings

Structured the graded Lie algebra of the fundamental group.

Extended Galois action understanding to all degrees of the lower central series.

Connected the algebraic structure to Galois representations on Fermat curves.

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 $\mathbb{Z}/n \mathbb{Z}$.