Monodromy of Fermat Surfaces and Modular Symbols for Fermat curves

Abstract

Let $F_n$ denote the Fermat curve given by $x^n+y^n=z^n$ and let ${\mu }_n$ denote the Galois module of $n$th roots of unity. It is known that the integral homology group $H_1(F_n,\mathbb{Z})$ is a cyclic $\mathbb{Z}[{\mu }_n\times {\mu }_n]$ module. In this paper, we prove this result using modular symbols and the modular description of Fermat curves; moreover we find a basis for the integral homology group $H_1(F_n,\mathbb{Z})$. We also construct a family of Fermat curves using the Fermat surface and compute its monodromy.

arXiv comment13 pages/accepted to Proceedings of AMS