The convergence of Eisenstein series of weight zero

2009-12-27T14:42:38Z

Consider Eisenstein series of weight zero, i.e.

$ E_{\mathfrak{a}}(z,\ s,\ \chi) = \sum_{ \gamma \in \Gamma_{\mathfrak{a}} \backslash \Gamma } \bar{\chi}(\gamma) (Im\sigma_{\mathfrak{a}}^{-1} \gamma z)^s $,

where $\chi$ is a multiplier system of weight zero ( $ \chi\ :\ \Gamma \rightarrow \mathbb{C}^* $ is a group homomorphism) singular at cusp $\mathfrak{a}$. Then my first question is that why this series converges absolutely in $Re(s)>1$?

My second question is how to calculate the following summation:

$ \sum_{d\ (mod c)}\ \epsilon_d(\frac{c}{d}) $, where $ \gamma = $ $[ \begin{pmatrix} a & b\ c & d \end{pmatrix} ]$ $\in \Gamma_0(4) $, $(\frac{c}{d})$ is the extended quadratic residue symbol and $c = b^2. $

Answer by Idoneal for The convergence of Eisenstein series of weight zero

2009-12-28T06:26:02Z

It seems you have started reading something from the middle.

Hint for the first one: Do it for $SL_2(\mathbb{Z})$ first. Note that

$Im \frac{az+b}{cz+d} = y/|cz+d|^2$.

The convergence of Eisenstein series of weight zero - MathOverflow

The convergence of Eisenstein series of weight zero

Answer by Idoneal for The convergence of Eisenstein series of weight zero