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)
[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)