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arXiv · 2001.11700

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

Siegfried Böcherer·Soumya Das
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Abstract

We study several aspects of nonvanishing Fourier coefficients of elliptic modular forms $\bmod \ell$, partially answering a question of Bella\"iche-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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Siegfried Böcherer, Soumya Das. 2020-01-31. On Fourier coefficients of elliptic modular forms $\bmod \, \ell$ with applications to Siegel modular forms. https://arxiv.org/abs/2001.11700

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