E

Eisenstein series One of the simplest examples of a modular form, defined as a sum over a lattice. In detail: Let be a discontinuous group of finite type operating on the upper half plane H , and let κ1, . . . , κh be a maximal set of cusps of which are not equivalent with respect to . Let i be the stabilizer in of κi , and fix an element σi ∈ G = SL(2,R) such that σi∞ = κi and such that σ−1i iσi is equal to the group 0 of all matrices of the form

( 1 b

)

with b ∈ Z. Denote by y(z) the imaginary part of z ∈ H . The Eisenstein series Ei(z, s) for the cusp κi is then defined by

Ei(z, s) = ∑

y(σ−1i σ z) s , σ ∈ 1 \ ,

where s is a complex variable.