For $f,g\in S_k(\Gamma(1))$, the <Petersson inner product> is
$$
\langle f,g\rangle
=\int_{\Gamma(1)\backslash\mathfrak h}
f(\tau)\overline{g(\tau)}y^k\frac{dx\,dy}{y^2}
=\int_{\mathcal F}f(\tau)\overline{g(\tau)}y^{k-2}\,dx\,dy.
$$
The transformation laws of $f$ and $g$, together with $\operatorname{Im}(\gamma\tau)=y/|c\tau+d|^2$, 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 $|x|\leq1/2$. Hence the absolute value of the integrand is
$$
O\left(y^{k-2}e^{-4\pi y}\right),
$$
which is integrable for large $y$. Therefore the Petersson integral converges absolutely.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-137/2/b/solution