# Modular group

The group $\Gamma$ of all fractional-linear transformations $\gamma$ of the form

$$\tag{1 } z \rightarrow \gamma ( z) = \ \frac{a z + b }{c z + d } ,\ \ a d - b c = 1 ,$$

where $a , b , c , d$ are rational integers. The modular group can be identified with the quotient group $\mathop{\rm SL} _ {2} ( \mathbf Z ) / \{ \pm E \}$, where

$$E = \left ( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right ) ,$$

and is a discrete subgroup in the Lie group $\mathop{\rm PSL} _ {2} ( \mathbf R ) = \mathop{\rm SL} _ {2} ( \mathbf R ) / \{ \pm E \}$. Here $\mathop{\rm SL} _ {2} ( \mathbf R )$( respectively, $\mathop{\rm SL} _ {2} ( \mathbf Z )$) is the group of matrices

$$\left ( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right ) ,$$

with $a , b , c , d$ real numbers (respectively, integers) and $ad - bc = 1$. The modular group is a discrete group of transformations of the complex upper half-plane $H = \{ {z = x + iy } : {y > 0 } \}$( sometimes called the Lobachevskii plane or Poincaré upper half-plane) and has a presentation with generators $T : z \rightarrow z + 1$ and $S : z \rightarrow - 1 / z$, and relations $S ^ {2} = ( ST) ^ {3} = 1$, that is, it is the free product of the cyclic group of order 2 generated by $S$ and the cyclic group of order 3 generated by $ST$( see ).

Interest in the modular group is related to the study of modular functions (cf. Modular function) whose Riemann surfaces (cf. Riemann surface) are quotient spaces of $H / \Gamma$, identified with a fundamental domain $G$ of the modular group. The compactification $X _ \Gamma = ( H / \Gamma ) \cup \infty$ is analytically isomorphic to the complex projective line, where the isomorphism is given by the fundamental modular function $J ( z)$. The fundamental domain $G$ has finite Lobachevskii area:

$$\int\limits _ { G } y ^ {-} 2 d x d y = \frac \pi {3} ,$$

that is, the modular group is a Fuchsian group of the first kind (see ). For the lattice $L = \mathbf Z + \mathbf Z z$, $z \in H$, the lattice $L _ {1} = \mathbf Z + \mathbf Z \gamma ( z)$,

$$\gamma = \ \left ( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right ) \in \Gamma ,$$

is equivalent to $L$, that is, can be obtained from $L$ by multiplying the elements of the latter by a non-zero complex number $\lambda$, $\lambda = ( c z + d ) ^ {-} 1$.

Corresponding to each lattice there is a complex torus $\mathbf C / L$ that is analytically equivalent to a non-singular cubic curve (an elliptic curve). This gives a one-to-one correspondence between the points of the quotient space $H / \Gamma$, classes of equivalent lattices and the classes of (analytically) equivalent elliptic curves (see ).

The investigation of the subgroups of the modular group is of interest in the theory of modular forms and algebraic curves (cf. Algebraic curve; Modular form). The principal congruence subgroup $\Gamma ( N)$ of level $N \geq 1$( $N$ an integer) is the group of transformations $\gamma ( z)$ of the form (1) for which $a \equiv d \equiv 1$( $\mathop{\rm mod} N$), $c \equiv b \equiv 0$( $\mathop{\rm mod} N$). A subgroup $\widetilde \Gamma \subset \Gamma$ is called a congruence subgroup if $\widetilde \Gamma \supset \Gamma ( N)$ for some $N$; the least such $N$ is called the level of $\widetilde \Gamma$. Examples of congruence subgroups of level $N$ are: the group $\Gamma _ {0} ( N)$ of transformations (1) with $c$ divisible by $N$, and the group $\Gamma _ {1} ( N)$ of transformations (1) with $a \equiv d \equiv 1$( $\mathop{\rm mod} N$) and $c \equiv 0$( $\mathop{\rm mod} N$). The index of $\Gamma ( N)$ in the modular group is $( N ^ {3} / 2 ) \prod _ {p \mid N } ( 1 - p ^ {-} 2 )$ if $N > 2$, $p$ is a prime number, and 6 if $N = 2$; thus, each congruence subgroup has finite index in the modular group.

Corresponding to each subgroup $\widetilde \Gamma$ of finite index in the modular group there is a complete algebraic curve $X _ {\widetilde \Gamma }$( a modular curve), obtained from the quotient space $H / \widetilde \Gamma$ and the covering $X _ {\widetilde \Gamma } \rightarrow X _ \Gamma$. The study of the branches of this covering allows one to find generators and relations for the congruence subgroup $\widetilde \Gamma$, the genus of $X _ {\widetilde \Gamma }$ and to prove that there are subgroups of finite index in the modular group which are not congruence subgroups (see , , , Vol. 2). The study of presentations of the modular group was initiated in work (see , ) connected with the theory of modular forms. Such presentations were intensively studied within the theory of automorphic forms (see  and Automorphic form). Many results related to the modular group have been transferred to the case of arithmetic subgroups of algebraic Lie groups (cf. Arithmetic group; Lie algebra, algebraic).

How to Cite This Entry:
Modular group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Modular_group&oldid=49312
This article was adapted from an original article by A.A. Panchishkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article