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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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<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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{{TEX|done}}
  
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
+
The group $  \Gamma $
 +
of all fractional-linear transformations  $  \gamma $
 +
of the form
  
<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>
+
$$ \tag{1 }
 +
z  \rightarrow  \gamma ( z)  = \
  
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
+
\frac{a z + b }{c z + d }
 +
,\ \
 +
a d - b c = 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/m06444010.png" /></td> </tr></table>
+
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
  
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]]]).
+
$$
 +
= \left (  
 +
\begin{array}{cc}
 +
1  & 0 \\
 +
0 & 1  \\
 +
\end{array}
 +
\right ) ,
 +
$$
  
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:
+
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 \} $.  
 +
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>
+
$$
 +
\left (
 +
\begin{array}{cc}
 +
a  & b  \\
 +
c  & d  \\
 +
\end{array}
 +
\right ) ,
 +
$$
  
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" />,
+
with  $  a , b , c , d $
 +
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]]]).
  
<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>
+
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 $,
 +
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" />.
+
$$
 +
\int\limits _ { G } y^{-2}  d x  d y  = \frac \pi {3}
 +
,
 +
$$
  
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]]]).
+
that is, the modular group is a [[Fuchsian group]] of the first kind (see [[#References|[3]]]). For the lattice  $  L = \mathbf Z + \mathbf Z z $,
 +
$  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.
+
$$
 +
\gamma  = \
 +
\left (  
 +
\begin{array}{cc}
 +
a & b  \\
 +
c  & d  \\
 +
\end{array}
 +
\right ) \in \Gamma ,
 +
$$
  
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]]).
+
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|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]]), 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]]; [[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) {{ZBL|0221.10029}}</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) {{ZBL|0376.10020}}</TD></TR>
 +
</table>

Latest revision as of 18:33, 13 January 2024


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=16537
This article was adapted from an original article by A.A. Panchishkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article