Modular group

The group of all fractional-linear transformations of the form

(1)

where are rational integers. The modular group can be identified with the quotient group , where

and is a discrete subgroup in the Lie group . Here (respectively, ) is the group of matrices

with real numbers (respectively, integers) and . The modular group is a discrete group of transformations of the complex upper half-plane (sometimes called the Lobachevskii plane or Poincaré upper half-plane) and has a presentation with generators and , and relations , that is, it is the free product of the cyclic group of order 2 generated by and the cyclic group of order 3 generated by (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 , identified with a fundamental domain of the modular group. The compactification is analytically isomorphic to the complex projective line, where the isomorphism is given by the fundamental modular function . The fundamental domain has finite Lobachevskii area:

that is, the modular group is a Fuchsian group of the first kind (see [3]). For the lattice , , the lattice ,

is equivalent to , that is, can be obtained from by multiplying the elements of the latter by a non-zero complex number , .

Corresponding to each lattice there is a complex torus 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 , 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 of level ( an integer) is the group of transformations of the form (1) for which (), (). A subgroup is called a congruence subgroup if for some ; the least such is called the level of . Examples of congruence subgroups of level are: the group of transformations (1) with divisible by , and the group of transformations (1) with () and (). The index of in the modular group is if , is a prime number, and 6 if ; thus, each congruence subgroup has finite index in the modular group.

Corresponding to each subgroup of finite index in the modular group there is a complete algebraic curve (a modular curve), obtained from the quotient space and the covering . The study of the branches of this covering allows one to find generators and relations for the congruence subgroup , the genus of 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)