The <Gamma 1 congruence subgroup> is
$$
\Gamma_1(N)=\left\{
\begin{pmatrix}a&b\\c&d\end{pmatrix}\in SL_2(\mathbb Z):
a\equiv d\equiv1\pmod N, c\equiv0\pmod N
\right\}.
$$
The space $M_k(\Gamma_1(N))$ consists of the <holomorphic functions> $f:\mathfrak h\to\mathbb C$ satisfying
$$
f(\gamma\tau)=(c\tau+d)^kf(\tau)
$$
for every $\gamma\in\Gamma_1(N)$ and which are <holomorphic at a cusp> for every cusp. Its subspace $S_k(\Gamma_1(N))$ consists of the forms that vanish at every cusp.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 137 / 2 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-137/2/a/solution