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

For $\tau =x+iy\in \mathfrak{h}$ and $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \Gamma (1)$,The set $\{c\tau +d:(c,d)\in {\mathbb{Z}}^2\setminus \{0\}\}$ is a lattice in $\mathbb{C}$, so it has a shortest nonzero vector. Dividing $(c,d)$ by their greatest common divisor can only shorten it; hence a minimizing pair may be chosen coprime and completed to the bottom row of some $\gamma \in S{L}_2(\mathbb{Z})$. Consequently the orbit contains a point ${\tau }_0$ of maximal imaginary part.

Apply a power of $T$ so that $-1/2\le \mathrm{Re}{\tau }_0\le 1/2$. If $|{\tau }_0|<1$, thencontradicting maximality. Thus $|{\tau }_0|\ge 1$, and ${\tau }_0$ lies in the standard fundamental domain of the modular group.

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.

Write $\sigma =\mathrm{Re}s$. The assumed coefficient bounds giveThe comparison p-series converges exactly whenor $\mathrm{Re}s>1+(k+l)/2$. Therefore the Rankin–Selberg convolution $L(f,g,s)$ converges absolutely in the stated half-plane.

Set $r=k-l$. Since $k$ and $l$ are even and $k\ge l+6$, one has $r\ge 6$. The holomorphic Eisenstein series in the question is absolutely convergent and decomposes asbecause every nonzero integer pair is a positive multiple of a primitive pair and the two signs contribute the factor two.

Absolute convergence, including that established in part (c) at $s=k-1$, permits Rankin–Selberg unfolding. Unfolding the Petersson inner product from the fundamental domain to the strip $0\le x<1$, $y>0$, givesThe $x$-integral uses orthogonality of complex exponentials to retain equal Fourier indices:Finally,Substitution gives the Rankin–Selberg unfolding identity for a holomorphic Eisenstein series