View source for Modular group

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

<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>

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

<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>

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

<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>

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]]]).

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:

<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>

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" />,

<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>

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" />.

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]]]).

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.

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]]).

====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>