 Modular function

A modular function for a finite-index subgroup $\Gamma \le S{L}_2(\mathbb{Z})$ is a meromorphic function on the complex upper half-plane that is invariant under $\Gamma$ and is meromorphic at every modular cusp:At a modular cusp of width $h$, meromorphicity means that the transformed function has a Laurent series in the local parameter $q_h=e^{2\pi i\tau /h}$ with only finitely many negative powers. Thus a modular function can have poles, in contrast to a modular form, which is required to be holomorphic on the half-plane and at the modular cusps. The Klein j-invariant is a fundamental example for the full modular group.

Some sources use the broader expression modular function of weight $k$ for a meromorphic function satisfyingwith the corresponding meromorphic cusp condition. This is also called a meromorphic modular form; the unqualified term modular function ordinarily means weight zero. Weight-zero modular functions descend to meromorphic functions on the compactified modular curve.^[ref]