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On Fourier coefficients of elliptic modular forms \(\bmod \, \ell \) with applications to Siegel modular forms

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Abstract

We study several aspects of nonvanishing Fourier coefficients of elliptic modular forms \(\bmod \, \ell \), partially answering a question of Bellaïche-Soundararajan concerning the asymptotic formula for the count of the number of Fourier coefficients upto x which do not vanish \(\bmod \, \ell \). We also propose a precise conjecture as a possible answer to this question. Further, we prove several results related to the nonvanishing of arithmetically interesting (e.g., primitive or fundamental) Fourier coefficients \(\bmod \, \ell \) of a Siegel modular form with integral algebraic Fourier coefficients provided \(\ell \) is large enough. We also make some efforts to make this “largeness” of \(\ell \) effective.

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Acknowledgements

S.D. was supported by a Humboldt Fellowship from the Alexander von Humboldt Foundation at Universität Mannheim during the preparation of the paper, and thanks both for the generous support and for providing excellent working conditions. The authors acknowledge the use of the LMFDB databse for some numerical computations. S.D also thanks IISc. Bangalore, DST (India) and UGC-CAS for financial support. During the preparation of this work S.D. was supported by a MATRICS grant MTR/2017/000496 from DST-SERB, India. We thank M. Raum for his comments on Siegel modular forms \(\bmod p\). We are thankful to the referee for a thorough checking of the paper and for giving valuable suggestions which improved the content.

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Authors and Affiliations

  1. Institut für Mathematik, Universität Mannheim, 68131, Mannheim, Germany

    Siegfried Böcherer

  2. Department of Mathematics, Indian Institute of Science, Bangalore, 560012, India

    Soumya Das

  3. Humboldt Fellow, Universität Mannheim, Mannheim, Germany

    Soumya Das

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  1. Siegfried Böcherer
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  2. Soumya Das
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Böcherer, S., Das, S. On Fourier coefficients of elliptic modular forms \(\bmod \, \ell \) with applications to Siegel modular forms. manuscripta math. 167, 405–434 (2022). https://doi.org/10.1007/s00229-021-01277-8

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