189-726B: Modular Forms II Assignment 8

Due: Wednesday, March 19

1. Let F be an imaginary quadratic field with ring of integers Z + Zω. Complete the proof started in class that

δ_{k}^{r}G_{k}(ω) = \frac{k(k + 1) ⋯ (k + r − 1)}{(ω − ω̄)^{r}}ζ_{F} (k + r, −r),

where

δ_{k}^{r} = δ_{k+2r−2} ⋯ δ_{k+2}δ_{k}

is the Shimura-Maass operator,

G_{k}(τ) = ∑^{′}(m + nτ)^{−k}

is the weight k Eisenstein series, and

ζ_{F} (m, n) = ∑_{α∈O_{F}} α^{−m}ᾱ^{−n}.