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

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.