Because $f$ is a modular function that is holomorphic on $\mathfrak h$, it has a Laurent expansion
$$
f(\tau)=\sum_{n\geq-N}a_nq^n
$$
with a finite principal part at infinity. The <Hecke operator on modular forms> acts on this expansion by
$$
T_pf=\sum_n\left(a_{pn}+p^{k-1}a_{n/p}\right)q^n.
$$

If $N>0$ and $a_{-N}\ne0$, the term $p^{k-1}f(p\tau)$ shows that $T_pf$ has pole order $pN$. Inductively, $T_p^rf$ has pole order $p^rN$ with nonzero leading coefficient. Functions with distinct pole orders are <linearly independent>, so
$$
f,T_pf,T_p^2f,\ldots
$$
would span an infinite-dimensional vector space. This contradicts the hypothesis. Hence $N=0$, and $f$ is holomorphic at infinity. Together with its assumed holomorphy on $\mathfrak h$, this proves that $f$ is a modular form, as asserted by the <finite Hecke orbit criterion for holomorphy at a cusp>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 137 / 2 / c / Solution

Let a level-one modular function be holomorphic on the upper half-plane. If the span of $f,T_pf,T_p^2f,\ldots$ is finite-dimensional, then $f$ is holomorphic at infinity. Indeed, a pole of order $N$ would make $T_p^rf$ have pole order $p^rN$, producing linearly independent functions.


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