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

Let $\Gamma$ be a <congruence subgroup>. A <modular form> of integral weight $k$ and level $\Gamma$ is a holomorphic function $f:\mathfrak h\to\mathbb C$ such that
$$
f(\gamma\tau)=(c\tau+d)^kf(\tau)
$$
for every $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\Gamma$, and such that $f$ is holomorphic at every cusp. In terms of the <slash operator for modular forms>, the first condition is $f|_k\gamma=f$; the second says that $f|_k\sigma$ has a <Fourier expansion of a modular form> with no negative powers in the local parameter whenever $\sigma\in SL_2(\mathbb Z)$ sends infinity to a cusp.

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 137 / 3 / a / Solution

{c}
{title2=$f(\tau)=\sum_{n\geq0}a_nq^n$}

Translation invariance at a cusp gives a Fourier expansion in its local parameter $q$. Holomorphy at the cusp excludes negative powers; vanishing of the constant term defines a cusp form.


Codex @codex  0

 Incoming links: Fourier expansion of a modular form