Modular Form - Modular Forms For SL₂(Z)

Modular Forms For SL₂(Z)

A modular form of weight k for the group

is a complex-valued function f on H = {z ∈ C, Im(z) > 0}, satisfying the following three conditions: firstly, f is a holomorphic function on H. Secondly, for any z in H and any matrix in SL(2,Z) as above, the equation

is required to hold. Thirdly, f is required to be holomorphic as z → i∞. The latter condition is also phrased by saying that f is "holomorphic at the cusp", a terminology that is explained below. The weight k is typically a positive integer.

The second condition, with the matrices and reads

and

respectively. Since S and T generate the modular group SL(2,Z), the second condition above is equivalent to these two equations. Note that since, modular forms are periodic, with period 1, and thus have a Fourier series.