Modular Forms and Hecke Operators

Question

I'm currently struggling to prove the following exercise:

Let $f:\mathbb{H}\to\mathbb{C}$ be holomorphic and $k\in\mathbb{N}$. If $f(\gamma z)=(cz+d)^{k}f(z)$ for every $\gamma\in Sl_{2}(\mathbb{Z})$ and $f$ is an eigenfunction of every Hecke Operator $T_{n}$, then $f$ is a modular form of weight $k$. Here $c$ and $d$ are given by

\begin{equation*} \gamma = \begin{pmatrix} a & b \newline c & d \end{pmatrix} \end{equation*}

For $f$ to be a modular form, all I need to show here is that $f$ is holomorphic at $\infty$. From my lecture notes I also know, that it is sufficient to show that there are constants $C,A>0$ such that $\lvert y^{\frac{k}{2}}f(x+iy)\rvert\leq C(y^{A}+y^{-A})$ for every $z=x+iy\in\mathbb{H}$.

However I don't see where to start. Any hint to the right direction is appreciated

Answer 1

That’s an interesting question!

I think the following approach works, but I’m not sure.

1. By mimicking the usual computations on modular forms, show that $(a_{-n}(f))_{n \geq 1}$ and $(a_n(f))_{n \geq 1}$ are proportional (and both are proportional to the sequence of Hecke eigenvalues $(\lambda_n)_{n \geq 1}$).
2. Show that $\sum_{n \geq 1}{a_{-n}(f)z^{n}}$ has infinite convergence radius.
3. Prove that, if $a_{-1}(f) \neq 0$, $\lambda_{p^n} = O(r^n)$ for every $r>0$.
4. Derive a contradiction with the Hecke recurrence relations. Deduce that $a_m(f)=0$ for all $m<0$.