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

Authors

Davis, R; Pries, R; Wickelgren, K

Abstract

Information about the absolute Galois group GK of a number field K is encoded in how it acts on the étale fundamental group π of a curve X defined over K. In the case that K = ℚ(ζn) is the cyclotomic field and X is the Fermat curve of degree n ≥ 3, Anderson determined the action of GK on the étale homology with coefficients in ℤ/nℤ. The étale homology is the first quotient in the lower central series of the étale fundamental group. In this paper, we determine the Galois module structure of the graded Lie algebra for π. As a consequence, this determines the action of GK 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 ℤ/nℤ.