发表机构
School of Mathematics and Statistics, Qingdao University（青岛大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过 Chebotarev--Sato--Tate 分布建立非负乘法函数在多元多项式取值处的对数上界，统一处理模形式等系数，并给出傅里叶系数的定量可除性结果。
AI 中文摘要
我们建立了非负乘法函数在多元多项式取值处的对数上界。多项式贡献由相应超曲面的几何不可约分量上的置换特征编码，而算术贡献则由联合 Chebotarev--Sato--Tate 群上的类函数描述。这为 CM 与非 CM 模形式的对称幂系数、Dedekind zeta 函数系数以及混合自守--伽罗瓦权提供了一个统一框架。作为进一步应用，我们得到了傅里叶系数沿多项式取值的定量可除性结果，包括在完全剩余像和 Eisenstein 同余情形下的显式公式。
英文摘要
We establish logarithmic upper bounds for nonnegative multiplicative functions evaluated at values of multivariable polynomials. The polynomial contribution is encoded by the permutation character on the geometric irreducible components of the corresponding hypersurface, while the arithmetic contribution is described by a class function on a joint Chebotarev--Sato--Tate group. This yields a unified framework for symmetric-power coefficients of CM and non-CM modular forms, Dedekind zeta-function coefficients, and mixed automorphic--Galois weights. As a further application, we obtain quantitative divisibility results for Fourier coefficients along polynomial values, including explicit formulas in the full residual-image and Eisenstein congruence cases.
