Modular curve

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

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

(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 ${\mathop{\rm mod}\nolimits} N$ 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 [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 $z \mathop \rightarrow \limits \gamma (z)$ of $\widetilde \Gamma$; here $f (z)$ is a cusp form of weight 2 relative to $\widetilde \Gamma$. The zeta-function 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 $E _ {\widetilde \Gamma } \mathop \rightarrow \limits X _ {\widetilde \Gamma }$ 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 [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 ${\mathop{\rm SL}\nolimits} _ {2} (A)$ 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, [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 $X_0(N)$ (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
[a1]	N.M. Katz, B. Mazur, "Arithmetic moduli of elliptic curves" , Princeton Univ. Press (1985)