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

A modular function of weight $k$ and level $\Gamma (1)$ is a meromorphic function $f:\mathfrak{h}\to \mathbb{C}$ satisfyingfor every $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \Gamma (1)$, and having a meromorphic Fourier expansion at the cusp infinity. Equivalently, $f{|}_k\gamma =f$ for the slash operator for modular forms.

A modular form is a modular function that is holomorphic on $\mathfrak{h}$ and holomorphic at infinity. Its Fourier expansion therefore has the formSince $-I\in \Gamma (1)$, a nonzero level-one form must have even weight.

Extend the slash operator for modular forms to positive-determinant matrices byThe double cosethas left-coset representativesTherefore the sum of the corresponding slashes, multiplied by $p^{k/2-1}$, is exactly

Right multiplication by an element of $\Gamma (1)$ permutes these left cosets. The cocycle law for the slash operator consequently givesso $T_p(f)$ is a weight-$k$ level-one modular function. Each displayed summand is holomorphic on $\mathfrak{h}$, hence so is their finite sum. This is the Hecke operator on modular forms.

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.