On Fourier coefficients of elliptic modular forms mod l with applications to Siegel modular forms

We study several aspects of nonvanishing Fourier coefficients of elliptic modular forms mod l, partially answering a question of Bellaiche-Soundararajan concerning the asymptotic formula for the count of the number of Fourier coefficients upto x which do not vanish mod l. 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 mod l of a Siegel modular form with integral algebraic Fourier coefficients provided l is large enough. We also make some efforts to make this "largeness" of l effective.