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.

我们研究了椭圆模形式\（\ bmod \，\ ell \）的不消失傅立叶系数的几个方面，部分回答了Bellaïche-Soundararajan的一个渐近公式的问题，该渐近公式用于计算直到不消失的x的傅立叶系数的数量\（\ bmod \，\ ell \）。我们还提出一个精确的猜想，作为对该问题的可能答案。此外，我们证明了一些结果，这些结果与具有积分代数傅里叶系数的Siegel模块化形式的Siegel模形式的算术有趣的（例如，原始的或基本的）傅立叶系数\（\ bmod \，\ ell \）消失不相关，前提是\（\ ell \）为足够大。我们还做出了一些努力，以使\（\ ell \）有效。