On the theory of Hilbert modular functions. I: Arithmetic groups and Eisenstein series
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 of such groups) and maximality of discrete groups of holomorphic automorphisms of the Cartesian product of upper half planes are studied in the next sections. The last section contains a discussion of Eisenstein series for the corresponding adèle group, which induce, on each component, holomorphic modular forms generalizing the familiar Eisenstein series due to Kloosterman. Arithmetic groups for which the Eisenstein series above generate the field of automorphic functions are realised more or less explicitly, viz. as group extensions of the usual arithmetic subgroups of \(PGL^+_ 2(k)\) by a (number-theoretically defined) subgroup of the group of automorphisms of k.
reciprocity laws for special values of Hilbert modular functions
maximality of discrete groups of holomorphic automorphisms
scientific article; zbMATH DE number 3875385
scientific article