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 mod , 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 mod . 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 of a Siegel modular form with integral algebraic Fourier coefficients provided is large enough. We also make some efforts to make this "largeness" of effective.

On Fundamental Fourier Coefficients of Siegel Modular Forms

“…At each stage we choose the smallest odd prime with these properties. As mentioned before, such primes are bounded solely in terms of κ and D (compare [11,Prop. 3.2]).…”

“…By multiplicity 1, we can choose an odd prime q = q 1,2 D such that b 1 (q) = b 2 (q). In fact, we can bound such a q solely in terms of κ and D (see, e.g., [11,Prop. 3.2]).…”