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ON FUNDAMENTAL FOURIER COEFFICIENTS OF SIEGEL MODULAR FORMS

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  03 March 2021

Siegfried Böcherer  and
Soumya Das Open the ORCID record for Soumya Das [Opens in a new window]
Siegfried Böcherer
Affiliation:
Institut für Mathematik, Universität Mannheim, 68131 Mannheim, Germany (boecherer@math.uni-mannheim.de)
Soumya Das
Affiliation:
Department of Mathematics, Indian Institute of Science, Bangalore – 560012, India (soumya@iisc.ac.in) Alexander von Humboldt Fellow, Universität Mannheim, 68131 Mannheim, Germany (sdas@mail.uni-mannheim.de)
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Abstract

We prove that if F is a nonzero (possibly noncuspidal) vector-valued Siegel modular form of any degree, then it has infinitely many nonzero Fourier coefficients which are indexed by half-integral matrices having odd, square-free (and thus fundamental) discriminant. The proof uses an induction argument in the setting of vector-valued modular forms. Further, as an application of a variant of our result and complementing the work of A. Pollack, we show how to obtain an unconditional proof of the functional equation of the spinor L-function of a holomorphic cuspidal Siegel eigenform of degree $3$ and level $1$ .

Keywords

Fourier coefficientsSiegel modular formsfundamental discriminantvector-valuednonvanishing

MSC classification

Primary: 11F30: Fourier coefficients of automorphic forms
Secondary: 11F50: Jacobi forms
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 6 , November 2022 , pp. 2001 - 2041
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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