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Difference between revisions of "Modular curve"

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A complete [[Algebraic curve|algebraic curve]] 
+
A complete [[algebraic curve]]    X _ {\widetilde \Gamma  } 
 
uniformized by a subgroup    \widetilde \Gamma 
 
uniformized by a subgroup    \widetilde \Gamma 
 
of finite index in the [[Modular group|modular group]]    \Gamma ;  
 
of finite index in the [[Modular group|modular group]]    \Gamma ;  
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The best known examples of subgroups    \widetilde \Gamma 
 
The best known examples of subgroups    \widetilde \Gamma 
 
of finite index in    \Gamma
 
of finite index in    \Gamma
are the congruence subgroups containing a principal congruence subgroup    \Gamma ( N)
+
are the [[congruence subgroup]]s containing a principal congruence subgroup    \Gamma ( N)
 
of level    N
 
of level    N
 
for some integer    N > 1 ,  
 
for some integer    N > 1 ,  
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$$  
 
$$  
A  \in  \mathop{\rm SL} _ {2} ( \mathbf Z ) ,\ \  
+
A  \in  \mathrm{SL} _ {2} ( \mathbf Z ) ,\ \  
 
A  \equiv \  
 
A  \equiv \  
 
\left (  
 
\left (  
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$$
 
$$
  
(see [[Modular group|Modular group]]). The least such    N
+
(see [[Modular group]]). The least such    N
 
is called the level of the subgroup    \widetilde \Gamma  .  
 
is called the level of the subgroup    \widetilde \Gamma  .  
 
In particular, the subgroup    \Gamma _ {0} ( N)
 
In particular, the subgroup    \Gamma _ {0} ( N)
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\prod _ {p \mid  N }
 
\prod _ {p \mid  N }
( 1 - p  ^ {-} 2 ) ,
+
( 1 - p  ^ {-2} ) ,
 
$$
 
$$
  
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does not contain    - 1 ,  
 
does not contain    - 1 ,  
 
compactified by degenerate curves above the parabolic points of    X _ {\widetilde \Gamma  }  .  
 
compactified by degenerate curves above the parabolic points of    X _ {\widetilde \Gamma  }  .  
Powers  $  E _ {down 2 {\widetilde \Gamma  }  } ^ {( w) } $,  
+
Powers    E _ { {\widetilde \Gamma  }  } ^ {( w) } ,  
 
where    w \geq  1
 
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) } $
+
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]]]).
 
are related to the Mellin transforms of modular forms, and their homology to the periods of modular forms (see [[#References|[3]]], [[#References|[7]]]).
  
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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  } 
 
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  ,  
 
with respect to decreasing    \widetilde \Gamma  ,  
which (in essence) coincides with the group  $  \mathop{\rm SL} _ {2} ( A) $
+
which (in essence) coincides with the group  $  \mathrm{ SL} _ {2} ( A) $
 
over the ring    A
 
over the ring    A
 
of rational adèles. On each modular curve    X _ {\widetilde \Gamma  } 
 
of rational adèles. On each modular curve    X _ {\widetilde \Gamma  } 

Revision as of 18:44, 14 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 \mathrm{SL} _ {2} ( \mathbf Z ) ,\ \ A \equiv \ \left ( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right ) \mathop{\rm mod} N

(see 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 [4], Vol. 2, [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 _ { {\widetilde \Gamma } } ^ {( w) } , where w \geq 1 is an integer, are called Kuga varieties (see [3], [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 [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 \mathrm{ 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=49734
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