Modular function - Definition and Overview

In mathematics, modular functions are certain kinds of mathematical functions mapping complex numbers to complex numbers. There are a number of other uses of the term "modular function" as well; see below for details.

Formally, a function j is called modular or a modular function iff it satisfies the following properties:

1. j is meromorphic in the upper half plane H.
2. For every matrix M in the modular group Γ, j(Mτ) = j(τ).
3. The Laurent series of j has the form

<math>j(\tau) = \sum_{n=-m}^\infty a(n) e^{2i\pi n\tau}.<math>

It can be shown that every modular function can be expressed as a rational function of Klein's absolute invariant J, and that every rational function of J is a modular function; furthermore, all modular functions are modular forms, although the converse does not hold. If a modular function j is not identically 0, then it can be shown that the number of zeroes of j is equal to the number of poles of j in the closure of the fundamental region R_Γ.

Other uses

There are a number of other usages of the term modular function, apart from this classical one; for example, in the theory of Haar measures, it is a function Δ(g) determined by the conjugation action.