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Mathematics > Number Theory

arXiv:1807.00391 (math)
[Submitted on 1 Jul 2018]

Title:Fourier expansions at cusps

Authors:François Brunault, Michael Neururer
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Abstract:In this article we study the number fields generated by the Fourier coefficients of modular forms at arbitrary cusps. We give upper bounds for these number fields which are cyclotomic extensions of the field generated by the Fourier coefficients at $\infty$, and determine them explicitly for newforms with trivial Nebentypus. The main tool is an extension of a result of Shimura on the compatibility between the actions of $\mathrm{SL}_2(\mathbb{Z})$ and $\mathrm{Aut}(\mathbb{C})$ on the space of modular forms. We give two new proofs of this result: one based on products of Eisenstein series, and the other using the theory of algebraic modular forms.
Subjects: Number Theory (math.NT)
Cite as: arXiv:1807.00391 [math.NT]
  (or arXiv:1807.00391v1 [math.NT] for this version)
  https://doi.org/10.48550/arXiv.1807.00391
Journal reference: Ramanujan J. 53 (2020), no. 2, 423-437
Related DOI: https://doi.org/10.1007/s11139-019-00178-5

Submission history

From: Michael Neururer [view email]
[v1] Sun, 1 Jul 2018 20:41:11 UTC (17 KB)
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