Eisenstein group cocycles for \(\text{GL}_ n\) and values of \(L\)- functions
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 \(\zeta({\mathfrak b},{\mathfrak f}; s)\) \((s=0,-1,\dots)\).
scientific article; zbMATH DE number 549667