The duality of cusp singularities (Q1204240)

The notion of cusp singularities on Hilbert modular surfaces was generalized in higher dimensions by Tsuchihashi by using the theory of toric varieties. In this paper, a dual relation of the invariants of the cusp singularities which was conjectured by \textit{I. Satake} and \textit{S. Ogata} [in Automorphic forms and geometry of arithmetic varieties, Adv. Stud. Pure Math. 15, 1-27 (1989; Zbl 0712.14009)] is proved. Namely, it is proved that the value at zero \(Z(0)\) of the zeta function associated to an even dimensional cusp singularity is equal to the arithmetic genus defect \(\chi_ \infty\) of its dual cusp singularity. This equality connects the dimension formula of the Hilbert modular cusp forms obtained by the Riemann-Roch theorem and that obtained by the Selberg trace formula.