AI 中文总结
该研究在分圆塔中建立ζ函数与模L值的同余关系，在GL₁、GL₂情形下分别推导相关同余并给出应用，为相关数论问题提供了新的研究路径。
AI 中文摘要
我们研究分圆塔中ζ函数与模L函数特殊值之间的同余关系，其核心机制是有限层Iwasawa理论元素的整范数同余，且该同余会随塔的层级升高而增强。在GL₁情形下，这给出了全实域上戴德金ζ值的同余关系，可应用于高阶K群、广义伯努利数及欧拉-庞加莱特征；在GL₂情形下，我们利用Mazur-Tate元素及其分布关系控制各分圆层级出现的本原扭特殊值，通过Artin形式主义，这些元素给出了连续分圆域上对应的基变换L值的同余关系。
英文摘要
We study congruences between special values of zeta functions and modular $L$-functions in cyclotomic towers. The underlying mechanism is an integral norm congruence for finite-level Iwasawa-theoretic elements, which becomes stronger as one ascends the tower. In the $\mathrm{GL}_1$ setting, this gives congruences for Dedekind zeta values over totally real fields, with applications to higher $K$-groups, generalized Bernoulli numbers, and Euler-Poincaré characteristics. In the $\mathrm{GL}_2$ setting, we use Mazur-Tate elements and their distribution relations to control the primitive twisted special values appearing at each cyclotomic level. Via Artin formalism, these give congruences for the corresponding base-change $L$-values over successive cyclotomic fields.
Commentsv1: 38 pages
