an:00059066
Zbl 0756.14001
Ishida, Masa-Nori
Cusp singularities given by reflections of stellable cones
EN
Int. J. Math. 2, No. 6, 635-657 (1991).
00002008
1991
j
14B05 14G10 14E15
Tsuchihashi cusps; arithmetic genus default; zeta zero
The author builds ``Tsuchihashi cusps'' [\textit{H. Tsuchihashi}, T??hoku Math. J., II. Ser. 35, 607-639 (1983; Zbl 0585.14004)] (this is a generalization of Hilbert modular cusp singularities). Such a singularity is defined by a pair \((C,\Gamma)\) of an open convex cone \(C\subset\mathbb{R}^ n\) and a discrete group \(\Gamma\subset GL(n,\mathbb{Z})\) with good conditions. The author defines and studies the notion of ``semi-integral stellable polyhedral cones'' \(C\), the group \(\Gamma\) generated by the reflections with respect to the facets of such a \(C\) gives rise to a good pair \((C,\Gamma)\). There is a duality among stellable cones, the corresponding singularities are dual in the sense of Tsuchihashi [loc. cit.].
At the end, the author gives effective examples of his singularities and computes the arithmetic genus default \(\chi_ \infty\) and the Ogata zeta zero \(Z(0)\) and verifies on his examples the Ogata-Satake conjecture: the \(\chi_ \infty\) of a cusp is equal to the \(Z(0)\) of its dual. A proof of this conjecture is announced as forthcoming.
[See also: \textit{E. B. Vinberg}, Math. USSR, Izv. 5(1971), 1083-1119 (1972); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 35, 1072-1112 (1971; Zbl 0247.20054) and \textit{T. Satake} and \textit{S. Ogata}, in Automorphic forms and geometry of arithmetic varieties, Adv. Stud. Pure Math. 15, 1-27 (1989; Zbl 0712.14009)].
V.Cossart (Versailles)
Zbl 0585.14004; Zbl 0256.20067; Zbl 0247.20054; Zbl 0712.14009