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The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m0644401.png" /> of all fractional-linear transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m0644402.png" /> of the form
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$#A+1 = 67 n = 0
 
$#C+1 = 67 : ~/encyclopedia/old_files/data/M064/M.0604440 Modular group
 
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m0644403.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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The group $  \Gamma $
+
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m0644404.png" /> are rational integers. The modular group can be identified with the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m0644405.png" />, where
of all fractional-linear transformations  $  \gamma $
 
of the form
 
  
$$ \tag{1 }
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m0644406.png" /></td> </tr></table>
z  \rightarrow  \gamma ( z)  = \
 
  
\frac{a z + b }{c z + d }
+
and is a [[Discrete subgroup|discrete subgroup]] in the [[Lie group|Lie group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m0644407.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m0644408.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m0644409.png" />) is the group of matrices
,\ \
 
a d - b c = 1 ,
 
$$
 
  
where  $  a , b , c , d $
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444010.png" /></td> </tr></table>
are rational integers. The modular group can be identified with the quotient group  $  \mathop{\rm SL} _ {2} ( \mathbf Z ) / \{ \pm  E \} $,
 
where
 
  
$$
+
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444011.png" /> real numbers (respectively, integers) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444012.png" />. The modular group is a [[Discrete group of transformations|discrete group of transformations]] of the complex upper half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444013.png" /> (sometimes called the Lobachevskii plane or Poincaré upper half-plane) and has a presentation with generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444015.png" />, and relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444016.png" />, that is, it is the free product of the cyclic group of order 2 generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444017.png" /> and the cyclic group of order 3 generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444018.png" /> (see [[#References|[2]]]).
= \left (
 
  
and is a [[Discrete subgroup|discrete subgroup]] in the [[Lie group|Lie group]] $  \mathop{\rm PSL} _ {2} ( \mathbf R ) = \mathop{\rm SL} _ {2} ( \mathbf R ) / \{ \pm  E \} $.  
+
Interest in the modular group is related to the study of modular functions (cf. [[Modular function|Modular function]]) whose Riemann surfaces (cf. [[Riemann surface|Riemann surface]]) are quotient spaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444019.png" />, identified with a fundamental domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444020.png" /> of the modular group. The compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444021.png" /> is analytically isomorphic to the complex projective line, where the isomorphism is given by the fundamental modular function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444022.png" />. The fundamental domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444023.png" /> has finite Lobachevskii area:
Here  $  \mathop{\rm SL} _ {2} ( \mathbf R ) $(
 
respectively, $  \mathop{\rm SL} _ {2} ( \mathbf Z ) $)
 
is the group of matrices
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444024.png" /></td> </tr></table>
\left (
 
  
with  $  a , b , c , d $
+
that is, the modular group is a [[Fuchsian group|Fuchsian group]] of the first kind (see [[#References|[3]]]). For the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444026.png" />, the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444027.png" />,
real numbers (respectively, integers) and  $  ad - bc = 1 $.
 
The modular group is a [[Discrete group of transformations|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 [[#References|[2]]]).
 
  
Interest in the modular group is related to the study of modular functions (cf. [[Modular function|Modular function]]) whose Riemann surfaces (cf. [[Riemann surface|Riemann surface]]) are quotient spaces of  $  H / \Gamma $,
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444028.png" /></td> </tr></table>
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:
 
  
$$
+
is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444029.png" />, that is, can be obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444030.png" /> by multiplying the elements of the latter by a non-zero complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444032.png" />.
\int\limits _ { G } y  ^ {-} 2  d x  d y  =
 
\frac \pi {3}
 
,
 
$$
 
  
that is, the modular group is a [[Fuchsian group|Fuchsian group]] of the first kind (see [[#References|[3]]]). For the lattice  $  L = \mathbf Z + \mathbf Z z $,
+
Corresponding to each lattice there is a complex torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444033.png" /> that is analytically equivalent to a non-singular cubic curve (an [[Elliptic curve|elliptic curve]]). This gives a one-to-one correspondence between the points of the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444034.png" />, classes of equivalent lattices and the classes of (analytically) equivalent elliptic curves (see [[#References|[3]]]).
$  z \in H $,
 
the lattice  $  L _ {1} = \mathbf Z + \mathbf Z \gamma ( z) $,
 
  
$$
+
The investigation of the subgroups of the modular group is of interest in the theory of modular forms and algebraic curves (cf. [[Algebraic curve|Algebraic curve]]; [[Modular form|Modular form]]). The principal congruence subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444036.png" /> of level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444037.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444038.png" /> an integer) is the group of transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444039.png" /> of the form (1) for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444040.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444041.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444042.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444043.png" />). A subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444044.png" /> is called a congruence subgroup if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444045.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444046.png" />; the least such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444047.png" /> is called the level of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444048.png" />. Examples of congruence subgroups of level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444049.png" /> are: the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444050.png" /> of transformations (1) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444051.png" /> divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444052.png" />, and the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444053.png" /> of transformations (1) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444054.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444055.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444056.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444057.png" />). The [[Index|index]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444058.png" /> in the modular group is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444059.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444061.png" /> is a prime number, and 6 if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444062.png" />; thus, each congruence subgroup has finite index in the modular group.
\gamma  = \
 
\left (
 
  
is equivalent to  $  L $,
+
Corresponding to each subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444063.png" /> of finite index in the modular group there is a complete algebraic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444064.png" /> (a [[Modular curve|modular curve]]), obtained from the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444065.png" /> and the covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444066.png" />. The study of the branches of this covering allows one to find generators and relations for the congruence subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444067.png" />, the genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064440/m06444068.png" /> and to prove that there are subgroups of finite index in the modular group which are not congruence subgroups (see [[#References|[3]]], [[#References|[8]]], [[#References|[7]]], Vol. 2). The study of presentations of the modular group was initiated in work (see [[#References|[4]]], [[#References|[6]]]) connected with the theory of modular forms. Such presentations were intensively studied within the theory of automorphic forms (see [[#References|[7]]] and [[Automorphic form|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|Arithmetic group]]; [[Lie algebra, algebraic|Lie algebra, algebraic]]).
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|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 [[#References|[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|Algebraic curve]]; [[Modular form|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|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|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 [[#References|[3]]], [[#References|[8]]], [[#References|[7]]], Vol. 2). The study of presentations of the modular group was initiated in work (see [[#References|[4]]], [[#References|[6]]]) connected with the theory of modular forms. Such presentations were intensively studied within the theory of automorphic forms (see [[#References|[7]]] and [[Automorphic form|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|Arithmetic group]]; [[Lie algebra, algebraic|Lie algebra, algebraic]]).
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Hurwitz,  R. Courant,  "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , '''1''' , Springer  (1964)  pp. Chapt.8</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.-P. Serre,  "A course in arithmetic" , Springer  (1973)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Shimura,  "Introduction to the arithmetic theory of automorphic functions" , Math. Soc. Japan  (1971)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E. Hecke,  "Analytische Arithmetik der positiven quadratischen Formen" , ''Mathematische Werke'' , Vandenhoeck &amp; Ruprecht  (1959)  pp. 789–918</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  F. Klein,  R. Fricke,  "Vorlesungen über die Theorie der elliptischen Modulfunktionen" , '''1–2''' , Teubner  (1890–1892)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  R.A. Rankin,  "Modular forms and functions" , Cambridge Univ. Press  (1977)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Hurwitz,  R. Courant,  "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , '''1''' , Springer  (1964)  pp. Chapt.8</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.-P. Serre,  "A course in arithmetic" , Springer  (1973)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Shimura,  "Introduction to the arithmetic theory of automorphic functions" , Math. Soc. Japan  (1971)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E. Hecke,  "Analytische Arithmetik der positiven quadratischen Formen" , ''Mathematische Werke'' , Vandenhoeck &amp; Ruprecht  (1959)  pp. 789–918</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  F. Klein,  R. Fricke,  "Vorlesungen über die Theorie der elliptischen Modulfunktionen" , '''1–2''' , Teubner  (1890–1892)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  R.A. Rankin,  "Modular forms and functions" , Cambridge Univ. Press  (1977)</TD></TR></table>

Revision as of 13:51, 7 June 2020

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)
How to Cite This Entry:
Modular group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Modular_group&oldid=49304
This article was adapted from an original article by A.A. Panchishkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article