On the theory of Hilbert modular functions. I: Arithmetic groups and Eisenstein series
type: Item
instance of: scholarly article
title: On the theory of Hilbert modular functions. I: Arithmetic groups and Eisenstein series
zbMATH Open document ID: 0549.10021
DOI: 10.1016/0021-8693(84)90190-X
author: Walter L. jun. Baily
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: The pioneering work of \textit{G. Shimura} [Ann. Math., II. Ser. 91, 144-222 (1970; Zbl 0237.14009), ibid. 92, 528-549 (1970; Zbl 0237.14010)] and Taniyama brought forth a far-reaching generalization of the classical theory of complex multiplication (and reciprocity laws for special values of the elliptic modular invariant), based on the theory of abelian varieties and their fields of moduli; further development resulted from Shimura's general theory of canonical models and a quite general reciprocity law for special values of modular functions of several variables on classical domains (of symplectic type). The object of this important paper - the first in a projected series of three papers - is to pave the way for a ''relatively elementary treatment of the reciprocity laws for special values of Hilbert modular functions'', by an investigation of the relevant arithmetic groups and Eisenstein series along with number-theoretic ramifications, with a view to obtain new results for these modular functions; these results are then applied to the ideas in Hecke's dissertation (and Habilitationsschrift), following also some ideas of Eichler and Hasse, in order eventually to ''recover'', in an ''elementary'' way (i.e. independent of abelian varieties etc.), some results pertaining to ''reciprocity laws for special values of arithmetic modular functions of several variables with respect to arithmetic groups acting on tube domains'' and certainly, in the first instance, for groups commensurable with the Hilbert modular group \(\Gamma\). Section 1 contains a list of necessary facts on central simple algebras over number fields k and Section 2, devoted to a description of maximal arithmetic subgroups of \(PGL_ 2(k)\), provides Helling's results (with alternative proofs for some) on the classification of conjugacy classes of maximal arithmetic groups commensurable with \(\Gamma\). ''Saturated'' arithmetic groups (with principal congruence subgroups constituting an important class…
Mathematics Subject Classification ID: 11F41
Mathematics Subject Classification ID: 11G15
Mathematics Subject Classification ID: 14G25
zbMATH DE Number: 3875385
zbMATH Keywords: complex multiplication
zbMATH Keywords: reciprocity laws for special values of Hilbert modular functions
zbMATH Keywords: arithmetic groups
zbMATH Keywords: Eisenstein series
zbMATH Keywords: maximal arithmetic groups
zbMATH Keywords: maximality of discrete groups of holomorphic automorphisms
zbMATH Keywords: adèle group
zbMATH Keywords: holomorphic modular forms
MaRDI profile type: Publication
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full work available at URL: 0021-8693(84)90190-x
OpenAlex ID: W1972680206
description: scientific article; zbMATH DE number 3875385