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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 \(n=2\).

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F75 Cohomology of arithmetic groups
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