Eisenstein series

Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generalized in the theory of automorphic forms.

Let $τ$ be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein series $G 2 k (τ)$ of weight $2 k$ , where $k \geq 2$ is an integer, by the following series:

This series absolutely converges to a holomorphic function of $τ$ in the upper half-plane and its Fourier expansion given below shows that it extends to a holomorphic function at $τ = i \infty$ . It is a remarkable fact that the Eisenstein series is a modular form. Indeed, the key property is its $SL(2, ℤ)$ -invariance. Explicitly if $a, b, c, d \in ℤ$ and $ad - bc = 1$ then

If $ad - bc = 1$ then

so that

is a bijection $ℤ 2 \to ℤ 2$ , i.e.:

Overall, if $ad - bc = 1$ then

and $G 2 k$ is therefore a modular form of weight $2 k$ . Note that it is important to assume that $k \geq 2$ , otherwise it would be illegitimate to change the order of summation, and the $SL(2, ℤ)$ -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 $k = 1$ , although it would only be a quasimodular form.

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