# 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

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

(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 , Vol. 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 ). 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 , ). 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 ).

A modular curve parametrizes a family of elliptic curves, being their moduli variety (see , 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 $E _ {down 2 {\widetilde \Gamma } } ^ {( w) }$, where $w \geq 1$ is an integer, are called Kuga varieties (see , ). The zeta- functions of $E _ {down 2 {\widetilde \Gamma } } ^ {( w) }$ are related to the Mellin transforms of modular forms, and their homology to the periods of modular forms (see , ).

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 ) 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 $\mathop{\rm SL} _ {2} ( A)$ 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, ).

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