论文标题
在椭圆模块的傅里叶系数上
On Fourier coefficients of elliptic modular forms $\bmod \, \ell$ with applications to Siegel modular forms
论文作者
论文摘要
我们研究了椭圆形模块化形式的非变化傅里叶系数的几个方面，部分回答了贝拉·奇奇·索拉拉杰恩（Bellaïche-Soundararajan）的问题，这些问题涉及渐近式公式，以计算傅立叶系数数量的数量，而不是$ x $，而不会消失\ bmod \ el el $ \ ell $。我们还提出了一个精确的猜想，以此作为对此问题的可能答案。此外，我们证明了几个与算术上有趣的（例如原始或基本的）傅立叶系数的划定相关的结果，该系数$ \ bmod \ ell $ $ \ ell $ $ \ ell $具有整体代数傅立叶系数，提供了$ \ ell $ $。我们还努力使这种“巨大”有效。
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.
