In §§23.15–23.19, $k$ and $k^{\prime }$ $(\in ℂ)$ denote the Jacobi modulus and complementary modulus, respectively, and $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$ ($\mathrm{\Im }\tau >0$) denotes the nome; compare §§20.1 and 22.1. Thus

Also $𝒜$ denotes a bilinear transformation on $\tau$, given by

in which $a,b,c,d$ are integers, with

The set of all bilinear transformations of this form is denoted by SL$(2,\mathbb{Z})$ (Serre (1973, p. 77)).

A modular function $f(\tau )$ is a function of $\tau$ that is meromorphic in the half-plane $\mathrm{\Im }\tau >0$, and has the property that for all $𝒜\in \text{SL}(2,\mathbb{Z})$, or for all $𝒜$ belonging to a subgroup of SL$(2,\mathbb{Z})$,

where $c_{𝒜}$ is a constant depending only on $𝒜$, and $\mathrm{\ell }$ (the level) is an integer or half an odd integer. (Some references refer to $2\mathrm{\ell }$ as the level). If, as a function of $q$, $f(\tau )$ is analytic at $q=0$, then $f(\tau )$ is called a modular form. If, in addition, $f(\tau )\to 0$ as $q\to 0$, then $f(\tau )$ is called a cusp form.