Fourier expansions at cusps
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Fourier expansions at cusps François Brunault1 · Michael Neururer2 Received: 23 November 2018 / Accepted: 13 May 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In this article, we study the fields generated by the Fourier coefficients of modular forms at arbitrary cusps. We prove that these fields are contained in certain cyclotomic extensions of the field generated by the Fourier coefficients at infinity. We also show that this bound is tight in the case of newforms with trivial Nebentypus. The main tool is a result of Shimura on the interplay between the actions of GL+ 2 (Q) and Aut(C) on modular forms. Keywords Modular forms · Fourier coefficients · Cusps · Cyclotomic extensions · Newforms · Atkin–Lehner operators Mathematics Subject Classification 11F11 · 11F30 · 11R18
1 Introduction In this article, we study the fields generated by the Fourier coefficients of modular forms at the cusps of X 1 (N ). To do this we study the connections between two actions on spaces of modular forms: the action of GL+ 2 (Q) via the slash-operator and the action of Aut(C) on the Fourier coefficients of a modular form. A detailed study of these actions was conducted by Shimura in [16], where he proved a formula for the action of Aut(C) on f |g. We provide a new proof of this result using a theorem of Khuri-Makdisi [12] on products of Eisenstein series. A second new proof from
The second author was partially funded by the DFG-Forschergruppe 1920 and the LOEWE research unit “Uniformized Structures in Arithmetic and Geometry”.
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Michael Neururer [email protected] François Brunault [email protected]
1
ÉNS Lyon, UMPA, 46 allée d’Italie, 69007 Lyon, France
2
TU Darmstadt, Schloßgartenstr. 7, 64289 Darmstadt, Germany
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F. Brunault, M. Neururer
the perspective of Katz’s theory of algebraic modular forms [11] is available on the arXiv [4]. We use this theorem to bound the fields generated by the Fourier coefficients of modular forms at the cusps. Let assume for simplicity that f is a modular form us B ∈ SL2 (Z). We show in Theorem 4.1 that the in Mk (0 (N )), and let g = CA D coefficients of f |g lie in the cyclotomic extension K f (ζ N ), where K f is the field generated by the coefficients of f , and N = N / gcd(C D, N ). In the case f has non-trivial Nebentypus, we show in Theorem 4.4 that the coefficients of f |g belong to a 1-dimensional K f (ζ N )-vector space, which is itself contained in an explicit cyclotomic extension K f (ζ M ). We apply these results in Sect. 5 to find number fields that contain the Atkin–Lehner pseudo-eigenvalues of a newform, recovering a result of Cohen in [5]. In Sect. 6, we discuss how to choose g among the matrices in SL2 (Z) that map ∞ to a given cusp α ∈ P1 (Q) so that N (or M) becomes minimal. Assuming that f ∈ Mk (0 (N )) is an eigenfunction of the Atkin–Lehner operators, we describe how to further reduce N by replacing α with its image under a suitable Atkin–Lehner operator. The Fourier expansion of f |g ca
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