Shintani–Barnes cocycles and values of the zeta functions of algebraic number fields

Abstract

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 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.