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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

$$  A  \in  {\mathop{\rm SL}\nolimits} _ {2} ( \mathbf Z ) ,\  A  \equiv  \left (  \begin{array}{cc}  1  & 0  \\  0  & 1  \\ \end{array}  \right )  \mathop{\rm mod}  N  $$

(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}\nolimits}  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  } \mathop \rightarrow \limits X _  \Gamma  $, 
which ramifies only over the images of the points  $  z = i $, 
$  z = ( 1 + i \sqrt 3 ) / 2 $, 
$  z = \inf $. 
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 [[#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 \mathop \rightarrow \limits \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  } \mathop \rightarrow \limits 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 _ { {\widetilde \Gamma  }} ^ {(w)} $, 
where  $  w \geq  1 $
is an integer, are called Kuga varieties (see [[#References|[3]]], [[#References|[5]]]). The zeta- functions of  $  E _ { {\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}\nolimits} _ {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====
<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====

====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>