Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 137 / 2

For fixed $\tau =x+iy$, the set $\{|c\tau +d|:(c,d)\in {\mathbb{Z}}^2\setminus \{0\}\}$ has a positive minimum. Choose a primitive pair $(c,d)$ attaining it and complete it to a matrix $\gamma \in S{L}_2(\mathbb{Z})$. Sincethis point has maximal imaginary part in the orbit. Translate by a power of $T:\tau \mapsto \tau +1$ to arrange $|\mathrm{Re}\tau |\le 1/2$. If now $|\tau |<1$, applying $S:\tau \mapsto -1/\tau$ strictly increases the imaginary part, a contradiction. Thus $|\tau |\ge 1$, proving that every orbit meets the standard fundamental domain of the modular group $\mathcal{F}$.

For a cusp form $f$, the invariant norm of a modular formis modular invariant, bounded on $\mathcal{F}$, and tends to zero at its cusp. Part a therefore makes it bounded throughout $\mathfrak{h}$: $|f(x+iy)|\le C{y}^{-k/2}$.

The correct PDF expansion is $f(\tau )={\sum }_{n\ge 1}{a}_n(f)q^n$. Fourier inversion givesand henceChoosing $y=1/n$ proves the Fourier coefficient bound for a cusp form $|a_n(f)|\le C^{\prime }{n}^{k/2}$.

The corrected PDF integrand isso the integral is a Petersson inner product. On compact subsets it is harmless, while at the cusp the exponential decay of $f$ dominates the polynomial growth of the Eisenstein series; therefore it converges absolutely.

Decompose each nonzero pair uniquely into a positive common divisor times a primitive pair. For even $k$ this writes $G_k$ as $2\zeta (k)$ times the Eisenstein sum over ${\Gamma }_{\infty }\backslash S{L}_2(\mathbb{Z})$. Unfolding the fundamental domain gives a constant multiple ofThe inner integral is the constant Fourier coefficient of the cusp form and is zero. Thus the original integral is zero, expressing the orthogonality of cusp forms and holomorphic Eisenstein series.