Modular Forms For SL2(Z)
A modular form of weight k for the group
is a complex-valued function f on H = {z ā 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 ā iā. 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.
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