Let $\Gamma \leq \text{SL}_2(\mathbb{Z})$ be a congruence subgroup and let $k \geq 0$ be an integer. Let $f: \mathbb{H} \rightarrow \mathbb{C}$ be a holomorphic function which is invariant by the slash-$k$-action of $\Gamma$:
$$ (f|_k \gamma)(z) = (cz +d)^{-k}f(\frac{a z + b}{cz + d}) = f(z) \quad \text{for all } \begin{pmatrix} a & b \\ c & d\end{pmatrix} \in \Gamma, z \in \mathbb{H}\,. $$
Define the function $F_f(z) := y^{k/2}|f(z)|$ for $z = x +iy \in \mathbb{H}$. Then $F_f$ is $\Gamma$-invariant: $F_f(\gamma \cdot z) = F_f(z)$ for all $z \in \mathbb{H}$ and all $\gamma \in \Gamma$.
For $\alpha \in\text{SL}_2(\mathbb{Z})$ the function $z \mapsto (f|_k \alpha)(z)$ is slash-$k$-invariant for the congruence (!) subgroup $\alpha^{-1}\Gamma \alpha$, hence is $h$-periodic for some integer $h > 0$. Using the map $q_h: z \mapsto e^{2 \pi i z /h}$ taking the upper-half plane to the punctured open unit disc $\mathbb{D}^{\ast}$, we obtain a holomorphic function $\tilde{f}_{\alpha} : \mathbb{D}^{\ast} \rightarrow \mathbb{C}$ such that $f(z) = \tilde{f}_{\alpha}(q(z))$ for $z \in \mathbb{H}$.
Question: Are the following equivalent?
- There exist constants $A, C \geq 0$ such that for all $z = x + iy \in \mathbb{H}$ one has $F_f(z) \leq C(y + y^{-1})^A$.
- For every $\alpha \in \text{SL}_2(\mathbb{Z})$,the function $\tilde{f}_{\alpha}$ extends to be holomorphic on the entire unit disc.
If yes, why? If not, what other growth conditions instead of the that given in 1. should one impose?
Similarly (for cusp forms), are the following equivalent?
- The function $F_f$ is bounded on the upper half plane.
- For every $\alpha \in \text{SL}_2(\mathbb{Z})$,the function $\tilde{f}_{\alpha}$ extends to be holomorphic on the entire unit disc and vanishes at zero.
(Of course, it suffices to check the second points for $\alpha$ running through a set of representatives of $\Gamma \backslash \text{SL}_2( \mathbb{Z})$.)