Hilbert’s Modular Group

Abstract

In a paper in Mathematische Annalen [13] Blumenthal did the first pioneering work in a program outlined by Hilbert with the aim of creating a theory of modular functions of several variables that should be just as important in number theory and geometry as the theory of modular functions of one variable was at the beginning of this century. Since no general theory of complex spaces was available this was by no means an easy task. Blumenthal had at his disposal a manuscript by Hilbert from 1893/94 on the action of the modular group ΓK of a totally real field K of degree n over ℚ on the product ℌn of n upper half planes. According to Blumenthal it gave a sketchy description of general properties such as properly discontinuous action and fundamental domain but it contained precise information on the construction of modular functions by means of theta functions. Blumenthal gave a detailed account of the function theory involved but his construction of a fundamental domain had a flaw: he obtained a fundamental domain with only one cusp as in the case of the classical modular group. This mistake was corrected many years later by Maass [110] who showed that the number of cusps equals the class number h of K.