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On the resolution of cusp singularities and the Shintani decomposition in totally real cubic number fields. (English) Zbl 0403.14005

Let \(K\) be a totally real number field of degree \(n\) and \(K\hookrightarrow \mathbb R^n\) the usual embedding. Let \(M\) be a lattice of rank \(n\) in \(K\) and \(V\) a subgroup of maximal rank \((n-1)\) of the group of those totally positive units which stabilize \(M\). Pairs \((M,V)\) describe cusps of the Hilbert modular variety associated to \(K\). In a beautiful series of papers, Hirzebruch has shown how to use the number theory of \(K\) to resolve these cusps when \(n=2\).
This papers gives completely explicit resolutions of these cusps for two infinite families of cubics. More generally, for any cubic it gives a fundamental domain, \(D\), for the action of \(V\) on \(\mathbb R^n_+\) such that \(D\) is a finite union of open simplicial cones with vertices in \(M\). Following M. Shintani [J. Fac. Sci., Univ. Tokyo, Sect. I A 23, 393–417 (1976; Zbl 0349.12007)], this can be used to calculate, as a finite sum, the values at nonnegative integers of a certain zeta function attached to \(M\). The methods used are geometric with a mild use of algebraic topology.
Reviewer: E. Thomas

MSC:

11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
14J17 Singularities of surfaces or higher-dimensional varieties
14J25 Special surfaces
14J30 \(3\)-folds
11R80 Totally real fields
11R16 Cubic and quartic extensions
14G25 Global ground fields in algebraic geometry

Citations:

Zbl 0349.12007
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References:

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