Eisenstein Series - Eisenstein Series For The Modular Group

Eisenstein Series For The Modular Group

Let be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein series of weight where is an integer, by the following series:

G_{2k}(\tau) = \sum_{ (m,n)\in\mathbb{Z}^2\backslash(0,0)} \frac{1}{(m+n\tau )^{2k}}.

This series absolutely converges to a holomorphic function of in the upper half-plane and its Fourier expansion given below shows that it extends to a holomorphic function at It is a remarkable fact that the Eisenstein series is a modular form. Indeed, the key property is its invariance. Explicitly if and then

G_{2k} \left( \frac{ a\tau +b}{ c\tau + d} \right) = (c\tau +d)^{2k} G_{2k}(\tau)

and is therefore a modular form of weight . Note that it is important to assume that otherwise it would be illegitimate to change the order of summation, and the -invariance would not hold. In fact, there are no nontrivial modular forms of weight 2. Nevertheless, an analogue of the holomorphic Eisenstein series can be defined even for although it would only be a quasimodular form.

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