Eisenstein group cocycles for \(\text{GL}_ n\) and values of \(L\)- functions
type: Item
instance of: scholarly article
title: Eisenstein group cocycles for \(\text{GL}_ n\) and values of \(L\)- functions
zbMATH Open document ID: 0809.11029
DOI: 10.1007/BF01244319
author: Robert Sczech
published in: Inventiones Mathematicae
publication date: 1995-04-02T00:00:00Z
full work available at URL: https://eudml.org/doc/144138
review text: Let \({\mathfrak b}\) and \({\mathfrak f}\) be two integral ideals in a number field \(F\) and denote by \(\zeta({\mathfrak b},{\mathfrak f}; s)\) \((s\in\mathbb{C})\) the zeta function of the ray class of \({\mathfrak b}\) modulo \({\mathfrak f}\). By work of Klingen-Siegel the special values \(\zeta({\mathfrak b},{\mathfrak f}; s)\) \((s=0,- 1, \dots)\) are known to be rational. In his paper the author gives an interpretation of these special values in terms of the cohomology of the group \(\Gamma= \text{GL}_ n (\mathbb{Z})\) \((n=\deg(F))\). More precisely, a so-called Eisenstein cocycle \(\psi\) on \(\Gamma\) is constructed which represents a non-trivial class in \(H^{n-1} (\Gamma, M)\) where \(M\) is a certain function space. Restricting \(\psi\) to the group \(U\) of totally positive units in \(F\) (embedded regularly in \(\Gamma\)) and evaluating functions in \(M\) on \(U\)-invariant points gives classes in \(H^{n-1} (U,\mathbb{C})\); among those certain rational classes when evaluated on a fundamental cocycle in \(H_{n-1} (U,\mathbb{Z})\), give the values \(\zeta({\mathfrak b},{\mathfrak f}; s)\) \((s=0,-1,\dots)\).
reviewed by: Winfried Kohnen
Mathematics Subject Classification ID: 11F67
zbMATH DE Number: 549667
zbMATH Keywords: zeta function
zbMATH Keywords: ray class
zbMATH Keywords: special values
zbMATH Keywords: cohomology
zbMATH Keywords: Eisenstein cocycle
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 549667