I am trying to solve the following exercise:
Let k $\in \mathbb {Z} $ and $f: \mathbb{H}\longrightarrow \mathbb {C} $ be a holomorphic function with $f \mid_k\gamma = f$ for all $\gamma \in\Gamma_1$ and $$f(x+iy) = O(1) (y \rightarrow+\infty).$$ show: for all $x\in \mathbb {Q} $ the following holds $$f(x+iy)= O(y^{(-k)}) (y\downarrow 0)$$ hint: show $f(\frac{a}{c}+iy)=({\frac{i}{c}})^k y^{-k} f(-\frac{d}{c}+\frac{i}{c^2y})$.
can you pls help me, i have approaches but I don't get continue
Let $f$ be a weight $k$ modular form for $SL(2,\mathbb Z)$. Let $\mathbb Q\ni x=\frac ac$ with $a,c\in\mathbb Z$, and $$\gamma=\begin{pmatrix}d&0\\ -c&a\end{pmatrix}\in SL(2,\mathbb Z).$$ In fact, $\gamma$ is just the inverse of $\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)$ with $b=0$. The relation $f|_k\gamma=f$ for $\tau=x+iy$ then reads $$f(\tfrac ac+i y)=\frac{1}{(-i c y)^k}f\left(-\tfrac dc+\tfrac{ ad}{c^2 y}i\right).$$ The limit $y\downarrow0$ of the second factor is a constant, $$\lim_{y\downarrow0}f\left(-\tfrac dc+\tfrac{i ad}{c^2 y}\right)=\lim_{y\uparrow\infty}f\left(-\tfrac dc+\tfrac{ ad y}{c^2 }i\right)=\mathcal O(1).$$ Therefore, the rhs diverges as $$\lim_{y\downarrow0}f(\tfrac ac+i y)=\mathcal O(y^{-k}).$$