In §§23.15–23.19, and denote the Jacobi modulus and complementary modulus, respectively, and () denotes the nome; compare §§20.1 and 22.1. Thus
Also denotes a bilinear transformation on , given by
in which are integers, with
The set of all bilinear transformations of this form is denoted by SL (Serre (1973, p. 77)).
A modular function is a function of that is meromorphic in the half-plane , and has the property that for all , or for all belonging to a subgroup of SL,
where is a constant depending only on , and (the level) is an integer or half an odd integer. (Some references refer to as the level). If, as a function of , is analytic at , then is called a modular form. If, in addition, as , then is called a cusp form.
compare also (23.15.2).
where (as in §20.2(i))