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

Abstract

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. Contents 1. Introduction 1 2. Preliminaries 3 3. The space Y • and the sheaves F d , F Ξ d 7 4. Equivariant cohomology 12 5. Cones and the exponential perturbation 20 6. Construction of the Shintani-Barnes cocycle 26 7. Integration 30 8. Specialization to the zeta values 36 References 48

Canonical equivariant cohomology classes generating zeta values of totally real fields

“…They mainly study certain group cocycles on GL n (Q) called the Eisenstein cocycles or the Shintani cocycles, while we stress a more algebrogeometric viewpoint by treating the geometric object T and equivariant sheaf cohomology classes. We also note that Bekki [7] studies L-values of general number fields by incorporating the ideas of both the above-mentioned works and this article.…”

“…We will now prove Theorem 5.1. Our assertion now follows from (7). In fact, we can modify the formulation to get rid of these dependences as follows.…”