Monodromy of Fermat Surfaces and Modular Symbols for Fermat curves

2016/10/10 by Ejder, Ozlem
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1610.02750

Abstract

Let F_n denote the Fermat curve given by xⁿ+yⁿ=zⁿ and let μ_n denote the Galois module of nth roots of unity. It is known that the integral homology group H₁(F_n,\Z) is a cyclic \Z[μ_n× μ_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 H₁(F_n,\Z). We also construct a family of Fermat curves using the Fermat surface and compute its monodromy.

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