Last edited by hbghlyj at 2025-1-29 13:04:00kuing.cjhb.site/forum.php?mod=viewthread&tid=10446 en.wikipedia.org/wiki/Modular_group
The modular group is the group of 2 × 2 matrices with integer coefficients and determinant 1. The matrices $A$ and $−A$ are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations. Presentation
The modular group can be shown to be generated by the two transformations\begin{aligned}S&:z\mapsto -{\frac {1}{z}}\\T&:z\mapsto z+1\end{aligned}so that every element in the modular group can be represented (in a non-unique way) by the composition of powers of $S$ and $T$. Geometrically, $S$ represents inversion in the unit circle followed by reflection with respect to the imaginary axis, while $T$ represents a unit translation to the right.
The generators $S$ and $T$ obey the relations $S^2 = 1$ and $(ST)^3 = 1$. It can be shown [1] that these are a complete set of relations, so the modular group has the presentation:$$ \Gamma \cong \left\langle S,T\mid S^{2}=I,\left(ST\right)^{3}=I\right\rangle $$
Modular Forms
Let $\mathbf{H}=\{\boldsymbol{\tau} \in \mathbf{C} \mid \operatorname{Im}(\tau)>0\}$ be the upper half plane, on which $\mathrm{SL}(2,\mathbf R)$ acts by the usual $(a \tau+b) /(c \tau+d)$-rule. The subgroup $\Gamma_{0}(N)$ of $\mathrm{SL}(2, \mathbf{Z})$ consists of those matrices
$$
\left(\begin{array}{ll}
a & b \\
c & d
\end{array}\right)
$$
with $c \equiv 0 \bmod N$. A modular form (of weight 2) for $\Gamma_{0}(N)$ is a holomorphic function $f(\tau)$ on $\bf H$ with
$$
f((a \tau+b) /(c \tau+d))=(c \tau+d)^{2} f(\tau)
$$
for
$$
\left(\begin{array}{ll}
a & b \\
c & d
\end{array}\right) \in \Gamma_{0}(N)
$$
and $f(\tau)$ "holomorphic at the cusps". This last statement means in particular for the Fourier series $($ since $f(\tau+1)=f(\tau))$
$$
f(\tau)=\sum_{n \in \mathbf{Z}} a_{n} \cdot e^{2 \pi i n \tau}
$$
that all $a_{n}$ vanish for $n<0$. If additionally $a_{0}=0$, then $f$ is called a cusp form. The Hecke algebra $\bf T$ acts on the space of cusp forms. It is generated by Hecke operators $T_{p}$ (for $p \nmid N$ prime) and $U_{p}$ (for $\left.p \mid N\right)$. For the Fourier coefficients one has\begin{aligned}
&a_{n}\left(T_{p} f\right)=a_{n p}(f)+p a_{n / p}(f), \\
&a_{n}\left(U_{p} f\right)=a_{n p}(f) .
\end{aligned}
