# Classical modular curve

In number theory, the **classical modular curve** is an irreducible plane algebraic curve given by an equation

- Φ
_{n}(*x*,*y*) = 0,

such that (*x*, *y*) = (*j*(*nτ*), *j*(*τ*)) is a point on the curve. Here *j*(*τ*) denotes the [[j-invariant|Template:Mvar-invariant]].

The curve is sometimes called *X*_{0}(*n*), though often that is used for the abstract algebraic curve for which there exist various models. A related object is the **classical modular polynomial**, a polynomial in one variable defined as Φ_{n}(*x*, *x*).

It is important to note that the classical modular curves are part of the larger theory of modular curves. In particular it has another expression as a compactified quotient of the complex upper half-plane **H**.

## Geometry of the modular curve

The classical modular curve, which we will call *X*_{0}(*n*), is of degree greater than or equal to 2*n* when *n* > 1, with equality if and only if Template:Mvar is a prime. The polynomial Φ_{n} has integer coefficients, and hence is defined over every field. However, the coefficients are sufficiently large that computational work with the curve can be difficult. As a polynomial in Template:Mvar with coefficients in **Z**[*y*], it has degree *ψ*(*n*), where Template:Mvar is the Dedekind psi function. Since Φ_{n}(*x*, *y*) = Φ_{n}(*y*, *x*), *X*_{0}(*n*) is symmetrical around the line *y* = *x*, and has singular points at the repeated roots of the classical modular polynomial, where it crosses itself in the complex plane. These are not the only singularities, and in particular when *n* > 2, there are two singularites at infinity, where *x* = 0, *y* = ∞ and *x* = ∞, *y* = 0, which have only one branch and hence have a knot invariant which is a true knot, and not just a link.

## Parametrization of the modular curve

For *n* = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, or 25, *X*_{0}(*n*) has genus zero, and hence can be parametrized [1] by rational functions. The simplest nontrivial example is *X*_{0}(2), where:

is (up to the constant term) the McKay–Thompson series for the class 2B of the Monster, and Template:Mvar is the Dedekind eta function, then

parametrizes *X*_{0}(2) in terms of rational functions of *j*_{2}. It is not necessary to actually compute *j*_{2} to use this parametrization; it can be taken as an arbitrary parameter.

## Mappings

A curve Template:Mvar, over **Q** is called a modular curve if for some Template:Mvar there exists a surjective morphism *φ* : *X*_{0}(*n*) → *C*, given by a rational map with integer coefficients. The famous modularity theorem tells us that all elliptic curves over **Q** are modular.

Mappings also arise in connection with *X*_{0}(*n*) since points on it correspond to Template:Mvar-isogenous pairs of elliptic curves. Two elliptic curves are *isogenous* if there is a morphism of varieties (defined by a rational map) between the curves which is also a group homomorphism, respecting the group law on the elliptic curves, and hence which sends the point at infinity (serving as the identity of the group law) to the point at infinity. The isogenies with cyclic kernel of degree Template:Mvar, the cyclic isogenies, correspond to points on *X*_{0}(*n*).

When *X*_{0}(*n*) has genus one, it will itself be isomorphic to an elliptic curve, which will have the same [[j-invariant|Template:Mvar-invariant]].

For instance, *X*_{0}(11) has Template:Mvar-invariant −2^{12}11^{−5}31^{3}, and is isomorphic to the curve *y*^{2} + *y* = *x*^{3} − *x*^{2} − 10*x* − 20. If we substitute this value of Template:Mvar for Template:Mvar in *X*_{0}(5), we obtain two rational roots and a factor of degree four. The two rational roots correspond to isomorphism classes of curves with rational coefficients which are 5-isogenous to the above curve, but not isomorphic, having a different function field. Specifically, we have the six rational points: x=-122023936/161051, y=-4096/11, x=-122023936/161051, y=-52893159101157376/11, and x=-4096/11, y=-52893159101157376/11, plus the three points exchanging Template:Mvar and Template:Mvar, all on *X*_{0}(5), corresponding to the six isogenies between these three curves.

If in the curve *y*^{2} + *y* = *x*^{3} − *x*^{2} − 10*x* − 20, isomorphic to *X*_{0}(11) we substitute

and factor, we get an extraneous factor of a rational function of Template:Mvar, and the curve *y*^{2} + *y* = *x*^{3} − *x*^{2}, with Template:Mvar-invariant −2^{12}11^{−1}. Hence both curves are modular of level 11, having mappings from *X*_{0}(11).

By a theorem of Henri Carayol, if an elliptic curve Template:Mvar is modular then its conductor, an isogeny invariant described originally in terms of cohomology, is the smallest integer Template:Mvar such that there exists a rational mapping *φ* : *X*_{0}(*n*) → *E*. Since we now know all elliptic curves over **Q** are modular, we also know that the conductor is simply the level Template:Mvar of its minimal modular parametrization.

## Galois theory of the modular curve

The Galois theory of the modular curve was investigated by Erich Hecke. Considered as a polynomial in x with coefficients in **Z**[*y*], the modular equation Φ_{0}(*n*) is a polynomial of degree *ψ*(*n*) in Template:Mvar, whose roots generate a Galois extension of **Q**(*y*). In the case of *X*_{0}(*p*) with Template:Mvar prime, where the characteristic of the field is not Template:Mvar, the Galois group of **Q**(*x*, *y*)/**Q**(*y*) is PGL(2, *p*), the projective general linear group of linear fractional transformations of the projective line of the field of Template:Mvar elements, which has *p* + 1 points, the degree of *X*_{0}(*p*).

This extension contains an algebraic extension *F*/**Q** where:

If we extend the field of constants to be Template:Mvar, we now have an extension with Galois group PSL(2, *p*), the projective special linear group of the field with Template:Mvar elements, which is a finite simple group. By specializing Template:Mvar to a specific field element, we can, outside of a thin set, obtain an infinity of examples of fields with Galois group PSL(2, *p*) over Template:Mvar, and PGL(2, *p*) over **Q**.

When Template:Mvar is not a prime, the Galois groups can be analyzed in terms of the factors of Template:Mvar as a wreath product.

## See also

## External links

## References

- Erich Hecke,
*Die eindeutige Bestimmung der Modulfunktionen q-ter Stufe durch algebraische Eigenschaften*, Math. Ann.**111**(1935), 293-301, reprinted in*Mathematische Werke*, third edition, Vandenhoeck & Ruprecht, Göttingen, 1983, 568-576 [3] - Anthony Knapp,
*Elliptic Curves*, Princeton, 1992 - Serge Lang,
*Elliptic Functions*, Addison-Wesley, 1973 - Goro Shimura,
*Introduction to the Arithmetic Theory of Automorphic Functions*, Princeton, 1972