Monodromy of Fermat Surfaces and Modular Symbols for Fermat curves

TL;DR

This paper investigates the monodromy of Fermat surfaces and describes the integral homology of Fermat curves using modular symbols, providing explicit bases and a modular perspective on their structure.

Contribution

It introduces a modular-symbol-based approach to analyze Fermat curves, explicitly constructs a basis for their homology, and explores the monodromy of Fermat surfaces.

Findings

Proved the cyclicity of $H_1(F_n,\Z)$ as a $\\\ ext{Z}[\mu_n \times \mu_n]$-module.

Constructed a basis for the integral homology group of Fermat curves.

Computed the monodromy of a family of Fermat surfaces.

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 $H_1(F_n,\Z)$ is a cyclic $\Z[\mu_n\times \mu_n]$ 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 $H_1(F_n,\Z)$. We also construct a family of Fermat curves using the Fermat surface and compute its monodromy.