Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 137 (Modular Forms)

The dimension of level-one cusp forms is zero when $k$ is odd or $k<12$. For even $k\ge 12$ it isEquivalently, multiplication by the modular discriminant gives $S_k(\Gamma (1))=\Delta M_{k-12}(\Gamma (1))$, and the displayed formula follows from the standard dimension formula for $M_{k-12}(\Gamma (1))$.

PutThe infinite product converges locally uniformly and never vanishes on the complex upper half-plane. With $D=qd/dq=(2\pi i{)}^{-1}d/d\tau$, logarithmic differentiation gives

The product is unchanged by $T:\tau \mapsto \tau +1$. To study $S:\tau \mapsto -1/\tau$, defineUsing the transformation law for the Eisenstein series of weight two,Thus $R$ is constant. At the fixed point $\tau =i$, one has $i^{12}=1$, so $R(i)=1$. HenceThe transformations under $S$ and $T$, which generate the modular group, show that $P$ is a weight-twelve modular form. Its Fourier expansion begins $q+O(q^2)$, so it is a cusp form. The normalized element of $S_{12}(\Gamma (1))$ is unique by part (a), and therefore

For $t>0$, the product from part (b) has $q=e^{-2\pi t}\in (0,1)$, soThe weight-twelve transformation law givesTogether with exponential decay as $t\to \infty$, this implies rapid decay at both endpoints for the Mellin transformThe integral therefore converges for every real $s$, and its integrand is strictly positive.

In the half-plane where the Dirichlet series may be integrated term by term,Analytic continuation preserves this identity. For real $s>0$, both $(2\pi {)}^s$ and the Gamma function $\Gamma (s)$ are positive, soThis is the positivity statement in the Mellin transform of the modular discriminant, and in particular $L(\Delta ,s)\ne 0$.

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

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 thatfor every $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \Gamma$, 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 $\sigma \in S{L}_2(\mathbb{Z})$ sends infinity to a cusp.

Let $\rho =e^{2\pi i/3}$. For a nonzero meromorphic modular form $f$ of weight $k$, the valence formula for the modular group isHere $v_z(f)$ is positive for a zero and negative for a pole, and the factors $1/2$ and $1/3$ account for the stabilizers of the two elliptic points. Equivalently, the weighted number of zeros minus poles in $\Gamma (1)\backslash \mathfrak{h}$ is $k/12-v_{\infty }(f)$.

The Poisson summation formula for the Gaussian gives the transformations of the Jacobi theta functionwhere the square root is the holomorphic branch on $\mathfrak{h}$. Raising to the $k$th power gives invariance under $T^2$. If $k=8m$, thenso ${\theta }^k(-1/\tau )={\tau }^{k/2}{\theta }^k(\tau )$. Since $T^2$ and $S$ generate the theta group, this is the weight-$k/2$ transformation law on $\Gamma$.

The defining series converges locally uniformly, so ${\theta }^k$ is holomorphic on $\mathfrak{h}$, and its expansion at infinity has no negative powers of $e^{\pi i\tau }$. Applying Poisson summation to the shifted Gaussian $e^{-\pi t(n+1/2{)}^2}$ gives the corresponding nonnegative-power expansion at the remaining cusp. Hence ${\theta }^k$ is holomorphic at every cusp and is a modular form of weight $k/2$ and level $\Gamma$.

The specialization of the Jacobi triple product to the Jacobi theta function isSince $\tau \in \mathfrak{h}$, one has $|q|<1$, so every displayed factor is nonzero. MoreoverThe standard convergence criterion for infinite products therefore shows that the product converges to a nonzero value. This proves the nonvanishing of the Jacobi theta function throughout $\mathfrak{h}$.

Let $\mathcal{L}$ be the set of lattices in $\mathbb{C}$. The lattice model of a modular form isFor the normalization used here, the $n$th Hecke operator on lattice functions isThere are finitely many such superlattices, and scaling gives a bijection between those above $\Lambda$ and those above $\lambda \Lambda$, so $T_{n}F$ again has weight $k$.

For $\tau \in \mathfrak{h}$, put ${\Lambda }_{\tau }=\mathbb{Z}\tau +\mathbb{Z}$ and defineIf $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \Gamma (1)$, thenso $f_F(\gamma \tau )=(c\tau +d{)}^k{f}_F(\tau )$. Conversely, if $f$ has this invariance and $\Lambda =\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2$ with $\mathrm{Im}({\omega }_1/{\omega }_2)>0$, defineThe modular transformation law makes this independent of the oriented basis. These constructions are inverse, identifying $V_k$ with $W_k$.

Transporting $T_n$ through this identification givesThis defines the Hecke operator on $W_k$.

Write $\tau =x+iy$ and $\sigma =\mathrm{Re}s>1$. The positive-definite quadratic formis bounded below by $C_{\tau }(m^2+n^2)$ for some $C_{\tau }>0$. Henceand the last two-dimensional lattice sum converges for $\sigma >1$. Thus the nonholomorphic Eisenstein series converges absolutely.

For $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \Gamma (1)$,andThe map $(m,n)\mapsto (ma+nc,mb+nd)$ permutes ${\mathbb{Z}}^2\setminus \{0\}$, so absolute convergence permits reindexing and gives $G(\gamma \tau ,s)=G(\tau ,s)$. Hence $G(\tau ,s)\in W_0$.

At weight zero the convention from part (a) isPartition the summands of $G$ according to divisibility of the first integer coordinate by $p$. Directly from the definition,In the sum over $b$, the congruence $n\equiv mb(\mathrm{mod}p)$ has one solution when $p\nmid m$, has $p$ solutions when $p$ divides both $m$ and $n$, and has none when $p\mid m$ but $p\nmid n$. Thereforewhere the contribution from pairs divisible by $p$ was rescaled by $p^{-2s}$. Dividing by $p$ proves the Hecke eigenvalue of a nonholomorphic Eisenstein series: