Congruence Subgroups
Congruence Subgroups
\[
\Gamma(N)=\{\gamma\in SL_2(\mathbb{Z})\mid \gamma\equiv 1_2\pmod N\}
\]
这里 $1_2$ 指二阶单位矩阵, 也即
\[
\Gamma(N)=\biggl\{
\begin{bmatrix} a & b\\ c & d\end{bmatrix}\in SL_2(\mathbb{Z})\biggl|\ a\equiv d\equiv 1,\ b\equiv c\equiv 0\ \mod N\mathbb{Z}
\biggr\}
\]
Congruence Subgroups 是指 $SL_2(\mathbb{Z})$ 中包含 $\Gamma(N)$ 的任一子群, $N$ 是某个自然数.
Math 252: Congruence Subgroups (wstein.org)
https://wstein.org/edu/Fall2003/252/lectures/09-19-03/index.html