The arithmetic genus of Hilbert modular varieties over non-Galois cubic fields
type: Item
instance of: scholarly article
title: The arithmetic genus of Hilbert modular varieties over non-Galois cubic fields
zbMATH Open document ID: 0715.11023
DOI: 10.1016/S0022-314X(05)80047-2
author: H. G. Grundman
published in: Journal of Number Theory
publication date: 1991-01-01T00:00:00Z
review text: It is well known that Hilbert modular varieties of totally real number fields of degree \(>6\) are of general type, so only in low degrees these varieties can be rational. On the other hand the arithmetic genus of such a variety always equals one. By the way, in low degrees too, almost all Hilbert modular varieties are of general type. In the paper under review the author elaborates for totally real cubic fields k the formula for the arithmetic genus due to Hirzebruch- Vignéras [\textit{F. Hirzebruch}, Enseignem. Math., II. Sér. 19, 183-281 (1973; Zbl 0285.14007), \textit{M.-F. Vignéras}, Math. Ann. 224, 189-215 (1976; Zbl 0325.12004)] in which the number of elliptic fixed points and \(\zeta_ k(-1)\) enter. Naturally, the calculations of the elliptic fixed points in terms of relative class numbers is based on the work of \textit{A. Prestel} [Math. Ann. 177, 181-209 (1968; Zbl 0159.113)] following the lines of \textit{D. Weisser} [Math. Ann. 257, 9-22 (1981; Zbl 0467.10021)] where these calculations have been done for Galois number fields. Concerning the relative class numbers and \(\zeta_ k(-1)\) \textit{T. Shintani}'s work [J. Fac. Sci., Univ. Tokyo Sect. IA 23, 393-417 (1976; Zbl 0349.12007)] on the evaluation of zeta-functions is used. Finally, numerical computations show that 24 of the 28 totally real non- Galois cubic fields of smallest discriminant are of arithmetic genus one thus contrasting the Galois case where just four Galois fields share this property.
Mathematics Subject Classification ID: 11F41
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R80
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 14G35
zbMATH DE Number: 4177166
zbMATH Keywords: Hirzebruch-Vignéras formula for the arithmetic genus
zbMATH Keywords: special 3-folds
zbMATH Keywords: Hilbert modular varieties
zbMATH Keywords: totally real cubic fields
zbMATH Keywords: elliptic fixed points
zbMATH Keywords: non-Galois cubic fields
zbMATH Keywords: arithmetic genus one
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4177166