page_title: Eisenstein series
section_title: Eisenstein series for the modular group
hierarchical_section_title: Eisenstein series / Eisenstein series for the modular group
context_page_description: 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.
context_section_description: Let τ be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein series G₂ₖ(τ) of weight 2k, where k ≥ 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∞. 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 ∈ ℤ and ad − bc = 1 then
  If ad − bc = 1 then
  so that
  is a bijection ℤ₂ → ℤ₂, i.e.:
  Overall, if ad − bc = 1 then
  and G₂ₖ is therefore a modular form of weight 2k. Note that it is important to assume that k ≥ 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.
caption: The real part of G6 as a function of q on the unit disk. Negative numbers are black.
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