Modular function: Difference between revisions

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 f is called modular or a modular function iff it satisfies the following properties:

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

$f(\tau )=\sum _{n=-m}^{\infty }a(n)e^{2i\pi n\tau }.$

It is bounded below; it is a Laurent polynomial in $e^{2i\pi \tau }$, so it is meromorphic at the cusp.

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

References

• Tom M. Apostol, Modular functions and Dirichlet Series in Number Theory (1990), Springer-Verlag, New York. ISBN 0-387-97127-0
• Robert A. Rankin, Modular forms and functions, (1977) Cambridge University Press, Cambridge. ISBN 0-521-21212-X