L-Series for Quadratic Forms at s = 1, II. The Modular Functions that Arise

Abstract

7. Modular functions. In this section necessary facts concerning the modular group and modular functions will be briefly summarized. More complete expositions of this material may be found in [3], [5], [6], [9], [15], [16], and [19]. Let r be the modular group, the group of all 2 X 2 matrices with rational integer entries and determinant equal to one. The elements S = (0 1) and T = (I 1) generate r. To an element A = (b b) E r, one associates a fractional linear transformation A(z) == (az + b)/(cz + d). The matrices A and -A give the same transformation. Let 3C* = 3C U {ioo} U Q where 3C is the upper half plane and Q is the set of rational points on the real axis. A fundamental domain is the set