Abstract: If X is a curve defined over a number field K, then we are motivated to understand the action of the absolute Galois group of K on the etale fundamental group of X. In this talk, I will focus on an important case, when X is the Fermat curve of degree p and K is the pth cyclotomic field. First, I will show how to reprove and extend some results of Greenberg about the p-torsion of the Jacobian of X using earlier work of Anderson, and of myself with Davis, Stojanoska, and Wickelgren. Then, I will discuss joint work with Duque-Rosero about the action of the absolute Galois group on the lower central series of the fundamental group, with coefficients mod p. The proofs involve some fun Galois theory and combinatorics.