# 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 [2]).

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 [3]). 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 [3]).

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 [3], [8], [7], Vol. 2). The study of presentations of the modular group was initiated in work (see [4], [6]) connected with the theory of modular forms. Such presentations were intensively studied within the theory of automorphic forms (see [7] 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).

#### References

 [1] A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , 1 , Springer (1964) pp. Chapt.8 [2] J.-P. Serre, "A course in arithmetic" , Springer (1973) (Translated from French) [3] G. Shimura, "Introduction to the arithmetic theory of automorphic functions" , Math. Soc. Japan (1971) Zbl 0221.10029 [4] E. Hecke, "Analytische Arithmetik der positiven quadratischen Formen" , Mathematische Werke , Vandenhoeck & Ruprecht (1959) pp. 789–918 [5] F. Klein, R. Fricke, "Vorlesungen über die Theorie der elliptischen Modulfunktionen" , 1–2 , Teubner (1890–1892) [6] H.D. Kloosterman, "The behaviour of general theta functions under the modular group and the characters of binary modular congruence groups I, II" Ann. of Math. , 47 (1946) pp. 317–375; 376–417 [7] J.-P. Serre (ed.) P. Deligne (ed.) W. Kuyk (ed.) , Modular functions of one variable. 1–6 , Lect. notes in math. , 320; 349; 350; 476; 601; 627 , Springer (1973–1977) [8] R.A. Rankin, "Modular forms and functions", Cambridge Univ. Press (1977) Zbl 0376.10020
How to Cite This Entry:
Modular group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Modular_group&oldid=55070
This article was adapted from an original article by A.A. Panchishkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article