Modular curve

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

(see Modular group). The least such is called the level of the subgroup . In particular, the subgroup represented by matrices which are congruent to upper-triangular matrices has level . Corresponding to each subgroup of finite index there is a covering of the modular curve , which ramifies only over the images of the points , , . For a congruence subgroup the ramification of this covering allows one to determine the genus of and to prove the existence of subgroups of finite index in which are not congruence subgroups (see [4], Vol. 2, [2]). The genus of is for and equals

a prime number, for . A modular curve is always defined over an algebraic number field (usually over 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 is given on by a differential (where is a holomorphic function) which is invariant under the transformations of ; here is a cusp form of weight 2 relative to . 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 (with conductor ) can be uniformized by modular functions of level . 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 -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 a point of is in one-to-one correspondence with a pair consisting of an elliptic curve (analytically equivalent to a complex torus ) and a point of order on (the image of ).

Over each modular curve there is a natural algebraic fibre bundle of elliptic curves if does not contain , compactified by degenerate curves above the parabolic points of . Powers , where is an integer, are called Kuga varieties (see [3], [5]). The zeta- functions of 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 .

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 with respect to decreasing , which (in essence) coincides with the group over the ring of rational adèles. On each modular curve this gives a non-trivial ring of correspondences (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