- The problem statement, all variables and given/known data
I need to show that
transforms with modular weight $2$ for $SL_2(Z)$
We have the theorem that it is sufficient to check the generators S and T
We have that E_2 is (whilst holomorphic) fails to transform with modular weight $2$ as it has this extra term when checking for $S$:
where transforming with weight $2$ means (for $S$):
$f(S.\tau)=\tau^{2}f(\tau)$
Therefore we expect a cancellation from the $Im (\tau)$ term
We have
MY QUESTION
- I follow
this solution and this is fine - I guess I am doing something pretty stupid, but we have the following formula
for $G=SL_2(R)$ and since $Z \in R$ we have $SL_2(Z) \in SL_2(R)$, so the above also holds for $SL_2(Z) $ and so this gives:
$Im(S.\tau)=\frac{Im(\tau)}{\tau^2}$
- so $1/Im(\tau)$ transforms with modular weight $2$ itself, and so is not cancelling
- I perhaps thought there may have been an issue that $\infty$ is not taken into account, but it also says the action on $G$ extends to $\infty$ , as I've said I follow the above solution, but have no idea what is wrong with using this formula.
Many thanks in advance