Difference between revisions of "Modular curve"
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− | A complete [[ | + | 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 ; | ||
Line 20: | Line 20: | ||
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 | + | 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 , | ||
Line 26: | Line 26: | ||
$$ | $$ | ||
− | A \in \ | + | A \in \mathrm{SL} _ {2} ( \mathbf Z ) ,\ \ |
A \equiv \ | A \equiv \ | ||
\left ( | \left ( | ||
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$$ | $$ | ||
− | (see [[ | + | (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) | ||
Line 61: | Line 61: | ||
\prod _ {p \mid N } | \prod _ {p \mid N } | ||
− | ( 1 - p ^ {-} | + | ( 1 - p ^ {-2} ) , |
$$ | $$ | ||
Line 95: | Line 95: | ||
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 _ { | + | 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 _ { | + | 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 $ \ | + | 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 ![]() |
[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) |
Modular curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Modular_curve&oldid=49734