Hi,


I've a question regarding the Petersson operator.


We have the following definition and the lemma.


Definition


Let $k, m \in \mathbb{Z}$ and $\phi: \mathbb{H} \times \mathbb{C} \rightarrow \mathbb{C}$. 


For


$\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \in \Gamma\_1$ und $\begin{pmatrix} \lambda & \mu \end{pmatrix} \in \mathbb{Z}^{2}$


we set


i) $\left( \phi|\_{k,m} \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \right) (\tau,z) :=
 (c\tau+d)^{-k} e^{2\pi i m \frac{-cz^2}{c\tau+d}} \phi(\frac{a\tau+b}{c\tau+d},\frac{z}{c\tau+d})$ `and


ii) $( \phi|\_m \begin{pmatrix} \lambda & \mu \\ \end{pmatrix}) (\tau,z) := 
 e^{2\pi i m (\lambda^2\tau + 2 \lambda z)} \phi(\tau, z + \lambda\tau+\mu) $


Lemma 


Let $k, m \in \mathbb{Z}$ and $\phi: \mathbb{H} \times \mathbb{C} \rightarrow \mathbb{C}$.


$M, M\_1, M\_2 \in \Gamma\_1;\; X, X\_1, X\_2 \in \mathbb{Z}^2$ satisfy


i) $(\phi|\_{k,m} M\_1)|\_{k,m} M\_2 = \phi|\_{k,m} (M\_1 M\_2)$


ii) $(\phi|\_m X\_1)|\_m X\_2 = \phi|\_m (X\_1 + X\_2)$


iii) $( \phi|\_{k,m} M)|\_m XM = (\phi|\_m X)|\_{k,m}M $


I've to prove that 


$ \phi|(M,X)|(M',X') = \phi|(MM',XM'+X')$


Unfortunately, after spending a lot of time, I am not clear how I can prove this. The main problem is that I don't know which calculation rule I can use. The lemma only deals with one variable ($M$ or $X$) but not with 2 variables $(M,X)$.


Do you have any hints how I could proceed.


Thanks and regards,
Jan



Modular form -  Petersson operator
I think the definition you want for $\phi|(M,X)$ is $(\phi |\_{k,m} M) |\_m X$.So your left hand side involving $\phi |(M,X) | (M',X')$ is the composition of four of these operators. Then the problem isn't too bad given your (i)-(iii).


