I'm new to modular form, reading the book A First Course in Modular Forms
We have the weight 2 Eisenstein series
$$
G_2(\tau)=\sum_{c\in\mathbb{Z}}\sum_{d\in\mathbb{Z}_c'}\frac{1}{(c\tau+d)^2}
$$
where $\mathbb{Z}_c'=\mathbb{Z}-\{0\}$ when $c=0$ otherwise $\mathbb{Z}$.
Now I am given that
$$
(G_2[\gamma]_2)(\tau)=G_2(\tau)-\frac{2\pi ic}{c\tau+d}\text{for }\gamma=\begin{bmatrix}a&b\\c&d\end{bmatrix}\in\text{SL}_2(\mathbb{Z})
$$
And I am asked to prove that if we know that the above formula is correct for two particular matrices $\gamma_1,\gamma_2$,then it is correct for $\gamma_1\gamma_2$.
I try to do as following:
$$
\begin{align*}
(G_2[\gamma_1\gamma_2]_2)(\tau)&=(G_2[\gamma_1]_2[\gamma_2]_2)(\tau)\text{ by property of the operator}\\
&=(G_2[\gamma_1]_2)(\gamma_2(\tau))\cdot j(\gamma_2,\tau)^{-1}\text{ by the definition}
\end{align*}
$$
But after substitute $\tau$ by $\gamma_2(\tau)$ in the above given formula, I can not get the desired equation.
Can anyone help?
Best Answer
Hint: Say $\gamma_1=\begin{pmatrix}a_1&b_1\\c_1&d_1\end{pmatrix}, \gamma_2=\begin{pmatrix}a_2&b_2\\c_2&d_2\end{pmatrix}, \gamma_1\gamma_2= \begin{pmatrix}a_3&b_3\\c_3&d_3\end{pmatrix}$, What you need to prove is that $[\gamma_1]_2({2\pi ic_2\over c_2\tau+d_2})+{2\pi ic_1\over c_1\tau+d_1}={2\pi c_3\over c_3\tau+d_3}$.
Expand the equation and verify it yourself, don't forget $\gamma\in SL_2$, that is $ad-bc=1$.