Special values of zeta functions associated to cusp singularities (Q1073119)

Let N be a free \({\mathbb{Z}}\)-module of rank n(\(>1)\) and \(N_{{\mathbb{R}}}:=N\otimes_{{\mathbb{Z}}}{\mathbb{R}}\). The author studies in this paper a zeta function Z(C,\(\Gamma\) ;s) associated to a nondegenerate open convex cone C in \(N_{{\mathbb{R}}}\) and to a subgroup \(\Gamma\) of \(Aut_{{\mathbb{Z}}}(N)\), such that C is \(\Gamma\)-invariant, \(\Gamma\) acts on \(D:=C/{\mathbb{R}}_{>0}\) property discontinuously and freely and D/\(\Gamma\) is compact. In the case that C is homogeneous and self dual, such functions where already considered by Shintani, Zagier and Satake. The author proves that Z(C,\(\Gamma\) ;s) can be continued meromorphically to the whole complex plane and, when n is odd, that the special value Z(C,\(\Gamma\) ;0) is equal to \(-2^{-1}e(D/\Gamma)\), where e(D/\(\Gamma)\) is the Euler number of the graph D/\(\Gamma\). In the odd case, this Euler number coincides with 2-times the cusp contribution \(\chi_{\infty}(p)\) of the Tsuchihashi cusp singularity of the cone. It is expected that the value Z(C,\(\Gamma\) ;0) should always be equal to \(-\chi_{\infty}(p)\).