For an integer $k$ and a <congruence subgroup> $\Gamma\leq SL_2(\mathbb Z)$, the space $M_k(\Gamma)$ consists of <holomorphic function>[holomorphic functions] $f:\mathfrak h\to\mathbb C$ satisfying
$$
(f|_k\gamma)(\tau)=(c\tau+d)^{-k}f(\gamma\tau)=f(\tau)
$$
for every $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\Gamma$, and which are <holomorphic at a cusp>[holomorphic at every cusp]. This is the space of <modular form>[modular forms] of weight $k$ and level $\Gamma$.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-137/1/a/solution