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Modular symbols for Fermat curves


Author: Özlem Ejder
Journal: Proc. Amer. Math. Soc. 147 (2019), 2305-2319
MSC (2010): Primary 11F23, 14D05, 11Gxx, 97F60
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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Additional Information

Özlem Ejder
Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523
Email: ejder@math.colostate.edu

DOI: https://doi.org/10.1090/proc/14396
Received by editor(s): March 20, 2018
Received by editor(s) in revised form: August 31, 2018, and September 13, 2018
Published electronically: March 5, 2019
Communicated by: Rachel Pries
Article copyright: © Copyright 2019 American Mathematical Society