Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 137 / 2 / b

For $f,g\in S_k(\Gamma (1))$, the Petersson inner product isThe transformation laws of $f$ and $g$, together with $\mathrm{Im}(\gamma \tau )=y/|c\tau +d{|}^2$, make the integrand invariant.

On every compact subset of $\mathcal{F}$ the integrand is bounded. At the only noncompact end, the Fourier expansion of a modular form and cuspidality give $f(\tau ),g(\tau )=O(e^{-2\pi y})$ uniformly for $|x|\le 1/2$. Hence the absolute value of the integrand iswhich is integrable for large $y$. Therefore the Petersson integral converges absolutely.