On the theory of Hilbert modular functions I. Arithmetic groups and Eisenstein series

Abstract

One of the objectives of this paper is to prepare for a relatively elementary treatment of the reciprocity laws for special values of Hilbert modular functions. Our attention was drawn to this subject by a paper of Eichler [ 151, in which a significant portion of the theory of complex multiplication and reciprocity laws for special values of the j-function was developed by elementary methods, and by a modified version of Eichler’s treatment due to Chevalley [lo]. Eichler’s paper caused us to look more carefully at Hecke’s thesis and Habilitationsschrift (Nos. 1 and 4 of his Mathematische Werke), where it was shown that the special values of arithmetic modular functions for the Hilbert modular group of a real quadratic number field generate abelian extensions of certain CM-fields, and a reciprocity law for some of these special values was established. Subsequently Sugawara [39] studied the fields generated by division values of suitably normalized abelian functions with complex multiplication. However, both of Hecke’s papers, though basically correct in their thrust, contained mistakes in function theory, and Sugawara’s work, though useful in some ways, was marred by deficiencies. Therefore, in the case of modular functions of several variables it remained for Shimura and Taniyama [37,40] to make pioneering breakthroughs in developing a general theory of complex multiplication based on the theory of abelian varieties and their fields of moduli. This was developed much further by Shimura [36] in providing a general theory of canonical models and deriving from it a very general reciprocity law for special values of modular functions of several variables on classical domains of symplectic type. After