Shintani-Barnes 上闭链与代数数域 zeta 函数的值
Shintani-Barnes cocycles and values of the zeta functions of algebraic number fields
AI总结：
本文通过引入指数扰动新技术，推广 Eisenstein 上闭链构造，建立了 Shintani-Barnes 上闭链，从而统一给出一般数域 zeta 函数在正整数处的值。
摘要：
本文构造了一个新的 Eisenstein 上闭链，称为 Shintani-Barnes 上闭链，它以统一的方式特殊化到一般数域的 zeta 函数在正整数处的值。我们的基本策略是推广 Vlasenko 和 Zagier 工作中给出的 Eisenstein 上闭链构造，利用 Bannai、Hagihara、Yamada 和 Yamamoto 在研究全实域多重对数函数时发展的一些近期技术。我们还紧密跟随 Charollois、Dasgupta 和 Greenberg 的工作。事实上，本文中使我们能够处理一般数域的关键要素之一，是引入了一种称为“指数扰动”的新技术，它是对他们工作中研究的 Q-扰动的一种轻微修改。
英文摘要：
In this paper, we construct a new Eisenstein cocycle called the Shintani-Barnes cocycle which specializes in a uniform way to the values of the zeta functions of general number fields at positive integers. Our basic strategy is to generalize the construction of the Eisenstein cocycle presented in the work of Vlasenko and Zagier by using some recent techniques developed by Bannai, Hagihara, Yamada, and Yamamoto in their study of the polylogarithm for totally real fields. We also closely follow the work of Charollois, Dasgupta, and Greenberg. In fact, one of the key ingredients in this paper which enables us to deal with general number fields is the introduction of a new technique called the ``exponential perturbation'' which is a slight modification of the Q-perturbation studied in their work.
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