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 → iāˆž. 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.

Read more about this topic:  Modular Form

Famous quotes containing the word forms:

    One way to think about play, is as the process of finding new combinations for known things—combinations that may yield new forms of expression, new inventions, new discoveries, and new solutions....It’s exactly what children’s play seems to be about and explains why so many people have come to think that children’s play is so important a part of childhood—and beyond.
    Fred Rogers (20th century)