Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 29 / 6 / c / ii

The transformation condition is necessary by the definition of a cusp form. For sufficiency, fix $\sigma \in S{L}_2(\mathbb{Z})$ representing any modular cusp, and set $g=f{|}_k\sigma$. By item (i), $g$ is holomorphic in the upper half-plane. Choose an actual period $h$ with $\sigma T^h{\sigma }^{-1}=\gamma \in \Gamma$. Associativity of slash operations and the given transformation law giveThus $g(\tau +h)=g(\tau )$ and $g$ is a single-valued holomorphic function on a punctured disk with coordinate $q_h=e^{2\pi i\tau /h}$. The calculation in part (b) transports the original uniform bound to this translate:This puts every modular cusp in exactly the situation needed in item (iii). No assumption that a single modular cusp represents all of $\Gamma$'s modular cusps is made.