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On Fourier coefficients of modular forms

On Fourier coefficients of modular forms By S. RAGI~VAN Let /(Z) be a modular form of degree n > 1 and weight k > 0 belonging to a principal congruence subgroup of stufe s of the Siegel modular group of degree n (see [2], [4]) and let /(g) =- ~. a(T) e ~Ltr(TZ) T>_O where T runs over n-rowed semi-integral non-negative matrices, be its Fourier expansion. If f(Z) is either an 'Eisenstein series' or a 'cusp form', there exists an estimate for the Fourier coefficients a(T) which involves the discriminant 5 (T) of T. Otherwise, for general ](Z), there exists for T ~ 0, an estimate for a(T) involving 5(T), only for k > n-~ 1 and only when T 'tends to infinity' in a special manner as described in [4]. The object of this note is to obtain an estimate for a(T) uniformly in T as the determinant of T tends to infinity and without requiring that k > n § 1. Since, e~en for subgroups of the Siegel modular group there exist for 0 < /c _< n § 1, n ~ 3 modular forms of weight/c, our theorem seems to be not without interest. The estimate that we obtain, however, is not http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

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References (8)

Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/BF02996934
Publisher site
See Article on Publisher Site

Abstract

By S. RAGI~VAN Let /(Z) be a modular form of degree n > 1 and weight k > 0 belonging to a principal congruence subgroup of stufe s of the Siegel modular group of degree n (see [2], [4]) and let /(g) =- ~. a(T) e ~Ltr(TZ) T>_O where T runs over n-rowed semi-integral non-negative matrices, be its Fourier expansion. If f(Z) is either an 'Eisenstein series' or a 'cusp form', there exists an estimate for the Fourier coefficients a(T) which involves the discriminant 5 (T) of T. Otherwise, for general ](Z), there exists for T ~ 0, an estimate for a(T) involving 5(T), only for k > n-~ 1 and only when T 'tends to infinity' in a special manner as described in [4]. The object of this note is to obtain an estimate for a(T) uniformly in T as the determinant of T tends to infinity and without requiring that k > n § 1. Since, e~en for subgroups of the Siegel modular group there exist for 0 < /c _< n § 1, n ~ 3 modular forms of weight/c, our theorem seems to be not without interest. The estimate that we obtain, however, is not

Journal

Abhandlungen aus dem Mathematischen Seminar der Universität HamburgSpringer Journals

Published: Dec 3, 2013

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