Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 137 / 1

For an integer $k$ and a congruence subgroup $\Gamma \le S{L}_2(\mathbb{Z})$, the space $M_k(\Gamma )$ consists of holomorphic functions $f:\mathfrak{h}\to \mathbb{C}$ satisfyingfor every $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \Gamma$, and which are holomorphic at every cusp. This is the space of modular forms of weight $k$ and level $\Gamma$.

For a compact subset $C\subset \mathfrak{h}$, the real-linear maphas inverse norm bounded uniformly for $\tau \in C$. Hence there is $A_C>0$ such thatTherefore the absolute value of the defining series is bounded locally uniformly bywhich converges for $k>2$. The Weierstrass M-test gives locally uniform absolute convergence, so termwise holomorphy proves that $G_k^{(x,y)}$ is holomorphic.

Write $\gamma =\left(\begin{array}{cc}a& b\\ r& s\end{array}\right)$. The automorphy factor identity givesRight multiplication by $\gamma$ bijects ${\mathbb{Z}}^2\setminus \{0\}$ and carries the congruence class $(x,y)$ to $(x,y)\gamma$. Reindexing the absolutely convergent series yields

If $\gamma \in \Gamma (N)$, then $(x,y)\gamma \equiv (x,y)(\mathrm{mod}N)$, so part c gives invariance. It remains to check the cusps. For any $\sigma \in S{L}_2(\mathbb{Z})$,As $\mathrm{Im}\tau \to \infty$, the terms with $c=0$ give a finite constant and the locally uniform estimate for the terms with $c\ne 0$ gives boundedness. A periodic holomorphic function bounded at infinity has a Fourier expansion with no negative powers. Thus every slash transform is holomorphic at infinity, which is holomorphy at every cusp. Hence $G_k^{(x,y)}$ is the congruence-class Eisenstein series in $M_k(\Gamma (N))$.