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A complete [[Algebraic curve|algebraic curve]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m0644101.png" /> uniformized by a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m0644102.png" /> of finite index in the [[Modular group|modular group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m0644103.png" />; more precisely, a modular curve is a complete algebraic curve obtained from a quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m0644104.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m0644105.png" /> is the upper half-plane, together with a finite number of parabolic points (the equivalence classes relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m0644106.png" /> of the rational points of the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m0644107.png" />). The best known examples of subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m0644108.png" /> of finite index in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m0644109.png" /> are the congruence subgroups containing a principal congruence subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441010.png" /> of level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441011.png" /> for some integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441012.png" />, represented by the matrices
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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/m064410/m06441013.png" /></td> </tr></table>
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{{TEX|done}}
  
(see [[Modular group|Modular group]]). The least such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441014.png" /> is called the level of the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441015.png" />. In particular, the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441016.png" /> represented by matrices which are congruent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441017.png" /> to upper-triangular matrices has level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441018.png" />. Corresponding to each subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441019.png" /> of finite index there is a covering of the modular curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441020.png" />, which ramifies only over the images of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441023.png" />. For a congruence subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441024.png" /> the ramification of this covering allows one to determine the genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441025.png" /> and to prove the existence of subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441026.png" /> of finite index in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441027.png" /> which are not congruence subgroups (see [[#References|[4]]], Vol. 2, [[#References|[2]]]). The genus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441028.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441029.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441030.png" /> and equals
+
A complete [[Algebraic curve|algebraic curve]]  $  X _ {\widetilde \Gamma  }  $
 +
uniformized by a subgroup  $  \widetilde \Gamma  $
 +
of finite index in the [[Modular group|modular group]] $  \Gamma $;
 +
more precisely, a modular curve is a complete algebraic curve obtained from a quotient space  $  H / \widetilde \Gamma  $,  
 +
where  $  H $
 +
is the upper half-plane, together with a finite number of parabolic points (the equivalence classes relative to  $  \widetilde \Gamma  $
 +
of the rational points of the boundary of $  H $).  
 +
The best known examples of subgroups $  \widetilde \Gamma  $
 +
of finite index in $  \Gamma $
 +
are the congruence subgroups containing a principal congruence subgroup  $  \Gamma ( N) $
 +
of level  $  N $
 +
for some integer  $  N > 1 $,
 +
represented by the 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/m064410/m06441031.png" /></td> </tr></table>
+
$$
 +
A  \in  \mathop{\rm SL} _ {2} ( \mathbf Z ) ,\ \
 +
A  \equiv \
 +
\left (
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441032.png" /> a prime number, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441033.png" />. A modular curve is always defined over an algebraic number field (usually over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441034.png" /> or a cyclic extension of it). The rational functions on a modular curve lift to modular functions (of a higher level) and form a field; the automorphisms of this field have been studied (see [[#References|[2]]]). A holomorphic differential form on a modular curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441035.png" /> is given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441036.png" /> by a differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441037.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441038.png" /> is a holomorphic function) which is invariant under the transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441039.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441040.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441041.png" /> is a cusp form of weight 2 relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441042.png" />. The [[Zeta-function|zeta-function]] of a modular curve is a product of the Mellin transforms (cf. [[Mellin transform|Mellin transform]]) of modular forms and, consequently, has a meromorphic continuation and satisfies a functional equation. This fact serves as the point of departure for the Langlands–Weil theory on the relationship between modular forms and Dirichlet series (see [[#References|[7]]], [[#References|[8]]]). In particular, there is a hypothesis that each [[Elliptic curve|elliptic curve]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441043.png" /> (with conductor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441044.png" />) can be uniformized by modular functions of level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441045.png" />. The homology of a modular curve is connected with modular symbols, which allows one to investigate the arithmetic of the values of the zeta-function of a modular curve in the centre of the critical strip and to construct the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441046.png" />-adic zeta-function of a modular curve (see [[#References|[1]]]).
+
(see [[Modular group|Modular group]]). The least such  $  N $
 +
is called the level of the subgroup  $  \widetilde \Gamma  $.  
 +
In particular, the subgroup  $  \Gamma _ {0} ( N) $
 +
represented by matrices which are congruent  $  \mathop{\rm mod}  N $
 +
to upper-triangular matrices has level  $  N $.  
 +
Corresponding to each subgroup  $  \widetilde \Gamma  $
 +
of finite index there is a covering of the modular curve  $  X _ {\widetilde \Gamma  }  \rightarrow X _  \Gamma  $,
 +
which ramifies only over the images of the points  $  z = i $,
 +
$  z = ( 1 + i \sqrt 3 ) / 2 $,
 +
$  z = \infty $.  
 +
For a congruence subgroup  $  \widetilde \Gamma  $
 +
the ramification of this covering allows one to determine the genus of $  X _ {\widetilde \Gamma  }  $
 +
and to prove the existence of subgroups  $  \widetilde \Gamma  $
 +
of finite index in  $  \Gamma $
 +
which are not congruence subgroups (see [[#References|[4]]], Vol. 2, [[#References|[2]]]). The genus of  $  X _ {\Gamma ( N) }  $
 +
is  $  0 $
 +
for  $  N \leq  2 $
 +
and equals
  
A modular curve parametrizes a family of elliptic curves, being their moduli variety (see [[#References|[7]]], Vol. 2). In particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441047.png" /> a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441048.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441049.png" /> is in one-to-one correspondence with a pair consisting of an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441050.png" /> (analytically equivalent to a complex torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441051.png" />) and a point of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441052.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441053.png" /> (the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441054.png" />).
+
$$
 +
1 +
  
Over each modular curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441055.png" /> there is a natural algebraic fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441056.png" /> of elliptic curves if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441057.png" /> does not contain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441058.png" />, compactified by degenerate curves above the parabolic points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441059.png" />. Powers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441060.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441061.png" /> is an integer, are called Kuga varieties (see [[#References|[3]]], [[#References|[5]]]). The zeta- functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441062.png" /> are related to the Mellin transforms of modular forms, and their homology to the periods of modular forms (see [[#References|[3]]], [[#References|[7]]]).
+
\frac{N  ^ {2} ( N - 6 ) }{24}
  
The rational points on a modular curve correspond to elliptic curves having rational points of finite order (or rational subgroups of points); their description (see [[#References|[6]]]) made it possible to solve the problem of determining the possible torsion subgroups of elliptic curves over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441063.png" />.
+
\prod _ {p \mid  N }
 +
( 1 - p  ^ {-} 2 ) ,
 +
$$
  
The investigation of the geometry and arithmetic of modular curves is based on the use of groups of automorphisms of the projective limit of the curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441064.png" /> with respect to decreasing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441065.png" />, which (in essence) coincides with the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441066.png" /> over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441067.png" /> of rational adèles. On each modular curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441068.png" /> this gives a non-trivial ring of correspondences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441069.png" /> (a Hecke ring), which has applications in the theory of modular forms (cf. [[Modular form|Modular form]], [[#References|[3]]]).
+
$  p $
 +
a prime number, for  $  N > 2 $.
 +
A modular curve is always defined over an algebraic number field (usually over  $  \mathbf Q $
 +
or a cyclic extension of it). The rational functions on a modular curve lift to modular functions (of a higher level) and form a field; the automorphisms of this field have been studied (see [[#References|[2]]]). A holomorphic differential form on a modular curve  $  X _ {\widetilde \Gamma  }  $
 +
is given on  $  H $
 +
by a differential  $  f ( z)  d z $(
 +
where  $  f ( z) $
 +
is a holomorphic function) which is invariant under the transformations  $  z \rightarrow \gamma ( z) $
 +
of  $  \widetilde \Gamma  $;
 +
here  $  f ( z) $
 +
is a cusp form of weight 2 relative to  $  \widetilde \Gamma  $.
 +
The [[Zeta-function|zeta-function]] of a modular curve is a product of the Mellin transforms (cf. [[Mellin transform|Mellin transform]]) of modular forms and, consequently, has a meromorphic continuation and satisfies a functional equation. This fact serves as the point of departure for the Langlands–Weil theory on the relationship between modular forms and Dirichlet series (see [[#References|[7]]], [[#References|[8]]]). In particular, there is a hypothesis that each [[Elliptic curve|elliptic curve]] over  $  \mathbf Q $(
 +
with conductor  $  N $)
 +
can be uniformized by modular functions of level  $  N $.
 +
The homology of a modular curve is connected with modular symbols, which allows one to investigate the arithmetic of the values of the zeta-function of a modular curve in the centre of the critical strip and to construct the  $  p $-
 +
adic zeta-function of a modular curve (see [[#References|[1]]]).
 +
 
 +
A modular curve parametrizes a family of elliptic curves, being their moduli variety (see [[#References|[7]]], Vol. 2). In particular, for  $  \widetilde \Gamma  = \Gamma ( N) $
 +
a point  $  z $
 +
of  $  H / \Gamma ( N) $
 +
is in one-to-one correspondence with a pair consisting of an elliptic curve  $  E _ {z} $(
 +
analytically equivalent to a complex torus  $  \mathbf C / ( \mathbf Z + \mathbf Z z) $)
 +
and a point of order  $  N $
 +
on  $  E _ {z} $(
 +
the image of  $  z / N $).
 +
 
 +
Over each modular curve  $  X _ {\widetilde \Gamma  }  $
 +
there is a natural algebraic fibre bundle  $  E _ {\widetilde \Gamma  }  \rightarrow X _ {\widetilde \Gamma  }  $
 +
of elliptic curves if  $  \widetilde \Gamma  $
 +
does not contain  $  - 1 $,
 +
compactified by degenerate curves above the parabolic points of  $  X _ {\widetilde \Gamma  }  $.
 +
Powers  $  E _ {down 2 {\widetilde \Gamma  }  } ^ {( w) } $,
 +
where  $  w \geq  1 $
 +
is an integer, are called Kuga varieties (see [[#References|[3]]], [[#References|[5]]]). The zeta- functions of  $  E _ {down 2 {\widetilde \Gamma  }  } ^ {( w) } $
 +
are related to the Mellin transforms of modular forms, and their homology to the periods of modular forms (see [[#References|[3]]], [[#References|[7]]]).
 +
 
 +
The rational points on a modular curve correspond to elliptic curves having rational points of finite order (or rational subgroups of points); their description (see [[#References|[6]]]) made it possible to solve the problem of determining the possible torsion subgroups of elliptic curves over  $  \mathbf Q $.
 +
 
 +
The investigation of the geometry and arithmetic of modular curves is based on the use of groups of automorphisms of the projective limit of the curves $  X _ {\widetilde \Gamma  }  $
 +
with respect to decreasing $  \widetilde \Gamma  $,  
 +
which (in essence) coincides with the group $  \mathop{\rm SL} _ {2} ( A) $
 +
over the ring $  A $
 +
of rational adèles. On each modular curve $  X _ {\widetilde \Gamma  }  $
 +
this gives a non-trivial ring of correspondences $  R _ {\widetilde \Gamma  }  $(
 +
a Hecke ring), which has applications in the theory of modular forms (cf. [[Modular form|Modular form]], [[#References|[3]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.I. Manin,  "Parabolic points and zeta-functions of modular curves"  ''Math. USSR Izv.'' , '''6''' :  1  (1972)  pp. 19–64  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''36''' :  1  (1972)  pp. 19–66</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Shimura,  "Introduction to the arithmetic theory of automorphic functions" , Math. Soc. Japan  (1971)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.V. [V.V. Shokurov] Šokurov,  "Holomorphic differential forms of higher degree on Kuga's modular varieties"  ''Math. USSR Sb.'' , '''30''' :  1  (1976)  pp. 119–142  ''Mat. Sb.'' , '''101''' :  1  (1976)  pp. 131–157</TD></TR><TR><TD valign="top">[4]</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">[5]</TD> <TD valign="top">  M. Kuga,  G. Shimura,  "On the zeta function of a fibre variety whose fibres are abelian varieties"  ''Ann. of Math.'' , '''82'''  (1965)  pp. 478–539</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B. Mazur,  J.-P. Serre,  "Points rationnels des courbes modulaires <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441070.png" /> (d'après A. Ogg)" , ''Sem. Bourbaki 1974/1975'' , ''Lect. notes in math.'' , '''514''' , Springer  (1976)  pp. 238–255</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">  A. Weil,  "Ueber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen"  ''Math. Ann.'' , '''168'''  (1967)  pp. 149–156</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.I. Manin,  "Parabolic points and zeta-functions of modular curves"  ''Math. USSR Izv.'' , '''6''' :  1  (1972)  pp. 19–64  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''36''' :  1  (1972)  pp. 19–66</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Shimura,  "Introduction to the arithmetic theory of automorphic functions" , Math. Soc. Japan  (1971)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.V. [V.V. Shokurov] Šokurov,  "Holomorphic differential forms of higher degree on Kuga's modular varieties"  ''Math. USSR Sb.'' , '''30''' :  1  (1976)  pp. 119–142  ''Mat. Sb.'' , '''101''' :  1  (1976)  pp. 131–157</TD></TR><TR><TD valign="top">[4]</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">[5]</TD> <TD valign="top">  M. Kuga,  G. Shimura,  "On the zeta function of a fibre variety whose fibres are abelian varieties"  ''Ann. of Math.'' , '''82'''  (1965)  pp. 478–539</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B. Mazur,  J.-P. Serre,  "Points rationnels des courbes modulaires <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064410/m06441070.png" /> (d'après A. Ogg)" , ''Sem. Bourbaki 1974/1975'' , ''Lect. notes in math.'' , '''514''' , Springer  (1976)  pp. 238–255</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">  A. Weil,  "Ueber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen"  ''Math. Ann.'' , '''168'''  (1967)  pp. 149–156</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.M. Katz,  B. Mazur,  "Arithmetic moduli of elliptic curves" , Princeton Univ. Press  (1985)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.M. Katz,  B. Mazur,  "Arithmetic moduli of elliptic curves" , Princeton Univ. Press  (1985)</TD></TR></table>

Revision as of 08:01, 6 June 2020


A complete algebraic curve $ X _ {\widetilde \Gamma } $ uniformized by a subgroup $ \widetilde \Gamma $ of finite index in the modular group $ \Gamma $; more precisely, a modular curve is a complete algebraic curve obtained from a quotient space $ H / \widetilde \Gamma $, where $ H $ is the upper half-plane, together with a finite number of parabolic points (the equivalence classes relative to $ \widetilde \Gamma $ of the rational points of the boundary of $ H $). The best known examples of subgroups $ \widetilde \Gamma $ of finite index in $ \Gamma $ are the congruence subgroups containing a principal congruence subgroup $ \Gamma ( N) $ of level $ N $ for some integer $ N > 1 $, represented by the matrices

$$ A \in \mathop{\rm SL} _ {2} ( \mathbf Z ) ,\ \ A \equiv \ \left ( (see [[Modular group|Modular group]]). The least such $ N $ is called the level of the subgroup $ \widetilde \Gamma $. In particular, the subgroup $ \Gamma _ {0} ( N) $ represented by matrices which are congruent $ \mathop{\rm mod} N $ to upper-triangular matrices has level $ N $. Corresponding to each subgroup $ \widetilde \Gamma $ of finite index there is a covering of the modular curve $ X _ {\widetilde \Gamma } \rightarrow X _ \Gamma $, which ramifies only over the images of the points $ z = i $, $ z = ( 1 + i \sqrt 3 ) / 2 $, $ z = \infty $. For a congruence subgroup $ \widetilde \Gamma $ the ramification of this covering allows one to determine the genus of $ X _ {\widetilde \Gamma } $ and to prove the existence of subgroups $ \widetilde \Gamma $ of finite index in $ \Gamma $ which are not congruence subgroups (see [[#References|[4]]], Vol. 2, [[#References|[2]]]). The genus of $ X _ {\Gamma ( N) } $ is $ 0 $ for $ N \leq 2 $ and equals $$ 1 +

\frac{N ^ {2} ( N - 6 ) }{24}

\prod _ {p \mid N } ( 1 - p ^ {-} 2 ) , $$

$ p $ a prime number, for $ N > 2 $. A modular curve is always defined over an algebraic number field (usually over $ \mathbf Q $ or a cyclic extension of it). The rational functions on a modular curve lift to modular functions (of a higher level) and form a field; the automorphisms of this field have been studied (see [2]). A holomorphic differential form on a modular curve $ X _ {\widetilde \Gamma } $ is given on $ H $ by a differential $ f ( z) d z $( where $ f ( z) $ is a holomorphic function) which is invariant under the transformations $ z \rightarrow \gamma ( z) $ of $ \widetilde \Gamma $; here $ f ( z) $ is a cusp form of weight 2 relative to $ \widetilde \Gamma $. The zeta-function of a modular curve is a product of the Mellin transforms (cf. Mellin transform) of modular forms and, consequently, has a meromorphic continuation and satisfies a functional equation. This fact serves as the point of departure for the Langlands–Weil theory on the relationship between modular forms and Dirichlet series (see [7], [8]). In particular, there is a hypothesis that each elliptic curve over $ \mathbf Q $( with conductor $ N $) can be uniformized by modular functions of level $ N $. The homology of a modular curve is connected with modular symbols, which allows one to investigate the arithmetic of the values of the zeta-function of a modular curve in the centre of the critical strip and to construct the $ p $- adic zeta-function of a modular curve (see [1]).

A modular curve parametrizes a family of elliptic curves, being their moduli variety (see [7], Vol. 2). In particular, for $ \widetilde \Gamma = \Gamma ( N) $ a point $ z $ of $ H / \Gamma ( N) $ is in one-to-one correspondence with a pair consisting of an elliptic curve $ E _ {z} $( analytically equivalent to a complex torus $ \mathbf C / ( \mathbf Z + \mathbf Z z) $) and a point of order $ N $ on $ E _ {z} $( the image of $ z / N $).

Over each modular curve $ X _ {\widetilde \Gamma } $ there is a natural algebraic fibre bundle $ E _ {\widetilde \Gamma } \rightarrow X _ {\widetilde \Gamma } $ of elliptic curves if $ \widetilde \Gamma $ does not contain $ - 1 $, compactified by degenerate curves above the parabolic points of $ X _ {\widetilde \Gamma } $. Powers $ E _ {down 2 {\widetilde \Gamma } } ^ {( w) } $, where $ w \geq 1 $ is an integer, are called Kuga varieties (see [3], [5]). The zeta- functions of $ E _ {down 2 {\widetilde \Gamma } } ^ {( w) } $ are related to the Mellin transforms of modular forms, and their homology to the periods of modular forms (see [3], [7]).

The rational points on a modular curve correspond to elliptic curves having rational points of finite order (or rational subgroups of points); their description (see [6]) made it possible to solve the problem of determining the possible torsion subgroups of elliptic curves over $ \mathbf Q $.

The investigation of the geometry and arithmetic of modular curves is based on the use of groups of automorphisms of the projective limit of the curves $ X _ {\widetilde \Gamma } $ with respect to decreasing $ \widetilde \Gamma $, which (in essence) coincides with the group $ \mathop{\rm SL} _ {2} ( A) $ over the ring $ A $ of rational adèles. On each modular curve $ X _ {\widetilde \Gamma } $ this gives a non-trivial ring of correspondences $ R _ {\widetilde \Gamma } $( a Hecke ring), which has applications in the theory of modular forms (cf. Modular form, [3]).

References

[1] Yu.I. Manin, "Parabolic points and zeta-functions of modular curves" Math. USSR Izv. , 6 : 1 (1972) pp. 19–64 Izv. Akad. Nauk SSSR Ser. Mat. , 36 : 1 (1972) pp. 19–66
[2] G. Shimura, "Introduction to the arithmetic theory of automorphic functions" , Math. Soc. Japan (1971)
[3] V.V. [V.V. Shokurov] Šokurov, "Holomorphic differential forms of higher degree on Kuga's modular varieties" Math. USSR Sb. , 30 : 1 (1976) pp. 119–142 Mat. Sb. , 101 : 1 (1976) pp. 131–157
[4] F. Klein, R. Fricke, "Vorlesungen über die Theorie der elliptischen Modulfunktionen" , 1–2 , Teubner (1890–1892)
[5] M. Kuga, G. Shimura, "On the zeta function of a fibre variety whose fibres are abelian varieties" Ann. of Math. , 82 (1965) pp. 478–539
[6] B. Mazur, J.-P. Serre, "Points rationnels des courbes modulaires (d'après A. Ogg)" , Sem. Bourbaki 1974/1975 , Lect. notes in math. , 514 , Springer (1976) pp. 238–255
[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] A. Weil, "Ueber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen" Math. Ann. , 168 (1967) pp. 149–156

Comments

References

[a1] N.M. Katz, B. Mazur, "Arithmetic moduli of elliptic curves" , Princeton Univ. Press (1985)
How to Cite This Entry:
Modular curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Modular_curve&oldid=13202
This article was adapted from an original article by A.A. PanchishkinA.N. Parshin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article