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

Because $f$ is a modular function that is holomorphic on $\mathfrak{h}$, it has a Laurent expansionwith a finite principal part at infinity. The Hecke operator on modular forms acts on this expansion by

If $N>0$ and $a_{-N}\ne 0$, the term $p^{k-1}f(p\tau )$ shows that $T_{p}f$ has pole order $pN$. Inductively, $T_p^{r}f$ has pole order $p^{r}N$ with nonzero leading coefficient. Functions with distinct pole orders are linearly independent, sowould 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.