Modular symbols for Fermat curves

Ejder, Özlem

doi:10.1090/proc/14396

Modular symbols for Fermat curves
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by Özlem Ejder

Proc. Amer. Math. Soc. 147 (2019), 2305-2319

DOI: https://doi.org/10.1090/proc/14396

Published electronically: March 5, 2019

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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.

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