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) |
Modular group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Modular_group&oldid=16537