# zbMATH — the first resource for mathematics

Shintani cocycles on $$\mathrm{GL}_n$$. (English) Zbl 1192.11030
Summary: The aim of this paper is to define an $$(n-1)$$-cocycle $$\sigma$$ on $$\mathrm{GL}_(\mathbb Q)$$ with values in a certain space $$\mathcal D$$ of distributions on $$\mathbb A_f^n\setminus \{0\}$$. Here $$\mathbb A_f$$ denotes the ring of finite adèles of $$\mathbb Q$$, and the distributions take values in the Laurent series $$\mathbb C((z_1,\dots,z_n))$$. This cocycle can be used to evaluate special values of Artin $$L$$-functions on number fields at negative integers. The construction generalizes that of D. Solomon [Compos. Math. 112, 333–362 (1998; Zbl 0920.11026), see also D. Solomon and H. Vu, Proc. Lond. Math. Soc. (3) 82, No. 1, 64–88 (2001; Zbl 1045.11035)] in the case $$n=2$$.

##### MSC:
 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11F75 Cohomology of arithmetic groups
Full Text: