Multi-tensors of differential forms on the Hilbert modular variety and on its subvarieties
type: Item
instance of: scholarly article
title: Multi-tensors of differential forms on the Hilbert modular variety and on its subvarieties
zbMATH Open document ID: 0576.10022
DOI: 10.1007/BF01458601
author: Shigeaki Tsuyumine
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164126
review text: It has been shown by \textit{E. Freitag} [Math. Z. 171, 27-35 (1980; Zbl 0445.10023)] and by \textit{G. Van der Geer} [On Hilbert modular surfaces of principal congruence subgroups, Dissertation, 1977] that Hilbert modular varieties for congruence subgroups of high level satisfy some minimality property, and moreover the automorphism groups of the modular function fields are determined. The author treats the problem in the case of level one. On the other hand, there is a conjecture that subvarieties of Hilbert modular varieties are of general type if they are not too special. The aim is done if it is proven that any subvariety in the Hilbert modular variety of codimension one is of general type. The author proves it for the Hilbert modular variety \(X_ K\) of a totally real algebraic number field K in the following cases; (i) almost all k of fixed degree \(n\geq 3\), and (ii) K of degree \(n\geq 40\) with 2 \(| n\), and (iii) K of degree n with \(2\nmid n\) such that \(8\pi \tilde n/| n-4\pi -1| <1\) where \(\tilde n=\prod p^{\ell (p)+1},\) \(n- 1=(p-1) p^{\ell (p)+1} m_ p,\) \(p\nmid m_ p\), p running over the set of primes with (p-1) \(| (n-1)\). So, under the same assumptions, the Hilbert modular variety \(X_ K\) satisfies some minimality property, and the automorphism group of the modular function field over \({\mathbb{C}}\) is isomorphic to a semi-direct product \(H_ K^{(2)}\rtimes Aut(K/{\mathbb{Q}})\), where \(H_ K^{(2)}=\{x\in H_ K|\) \(x^ 2=1\}\), \(H_ K\) denoting the class group of K in the narrow sense. His method is to show the existence of enough multi-tensors of differentials extendable to a projective non-singular model of \(X_ K\) whose restrictions to any subvariety in \(X_ K\) of codimension one gives many pluri-canonical differentials on it, where \textit{K. B. Gundlach}'s estimate of weights of automorphy factors for the Hilbert modular group [Glasgow Math. J. 27, 57-80 (1985; see the preceding review)] plays an important role.
Mathematics Subject Classification ID: 11F41
Mathematics Subject Classification ID: 14J20
zbMATH DE Number: 3920581
zbMATH Keywords: congruence subgroups
zbMATH Keywords: level one
zbMATH Keywords: subvarieties of general type
zbMATH Keywords: Hilbert modular variety
zbMATH Keywords: minimality property
zbMATH Keywords: automorphism group
zbMATH Keywords: modular function field
zbMATH Keywords: weights of automorphy factors
zbMATH Keywords: Hilbert modular group
MaRDI profile type: Publication
cites work: Q4099744
cites work: Lokale und globale Invarianten der Hilbertschen Modulgruppe
cites work: Eine Bemerkung zur Theorie der Hilbertschen Modulmannigfaltigkeiten hoher Stufe
cites work: Q4119324
cites work: Multiplier systems for Hilbert's and Siegel's modular groups
cites work: Hilbert modular surfaces and the classification of algebraic surfaces
cites work: Q5612629
cites work: Die elliptischen Fixpunkte der Hilbertschen Modulgruppen
cites work: Q4154638
cites work: Le problème des groupes de congruence pour \(SL_2\)
cites work: On discontinuous groups operating on the product of the upper half planes
cites work: On the Kodaira dimension of the moduli space of Abelian varieties
cites work: On the Kodaira dimensions of Hilbert modular varieties
cites work: Multi-tensors of differential forms on the Siegel modular variety and on its subvarieties
OpenAlex ID: W1998875784
description: scientific article; zbMATH DE number 3920581