Classical modular curve: Difference between revisions

Revision as of 22:27, 28 May 2006

Revision as of 09:11, 18 July 2006

Line 1:

In [[number theory]], the '''classical modular curve''' is an irreducible [[algebraic curve|plane algebraic curve]] given by an equation

:ΦΦ_n(x, y)=0,

where for the [[j-invariant]] j(ττ),

:x=j(n ττ), y=j(ττ)

is a point on the curve. The curve is sometimes called X₀(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 ΦΦ_n(x, x).

== Geometry of the modular curve ==

[[Image:Modknot11.png|thumb|Knot at infinity of X₀(11)]]

The classical modular curve, which we will call X₀(n), is of degree greater than or equal to 2n when n>1, with equality if and only if n is a prime. The polynomial ΦΦ_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 x with coefficients in '''Z'''[y], it has degree ψψ(n), where ψψ is the [[Dedekind psi function]]. Since ΦΦ_n(x, y) =    ΦΦ_n(y, x), X₀(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 ==

Line 27:

== Mappings ==

A curve C over the rationals '''Q''' such that there exists a surjective morphism from X<sub>0</sup>(n) to C for some n, given by a rational map with integer coefficients &phi;φ:X<sub>0</sup>(n) → C, is a [[modular curve]]. The famous [[modularity theorem]] tells us that all [[elliptic curve]]s over '''Q''' are modular.

Mappings also arise in connection with X<sub>0</sup>(n) since points on it correspond to n-isogenous pairs of elliptic curves. Two elliptic curves are ''isogenous'' if there is a morphism of varitiesvarieties (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 n, the cyclic isogenies, correspond to points on X<sub>0</sup>(n).

When X<sub>0</sup>(n) has genus one, it will itself be isomorphic to an elliptic curve, which will have the same [[j-invariant]]. For instance, X<sub>0</sup>(11) has j-invariant -122023936/161051 = - 2¹²11^-531³, and is isomorphic to the curve y²+y = x³-x<sup>2^{-10x-20. If we substitute this value of j for y in X<sub>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.

Line 40:

By a theorem of [[Henri Carayol]], if an elliptic curve E is modular then its conductor, an isogeny invariant described originally in terms of [[cohomology]],

is the smallest integer n such that there exists a rational mapping &phi;φ:X₀(n)</sub> → E. Since we now know all elliptic curves over '''Q''' are modular, we also know that the conductor is simply the level n 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 &Phi;Φ₀(n) is a polynomial of degree &psi;ψ(n) in x, whose roots generate a [[Galois extension]] of '''Q'''(y). In the case of X₀(p) with p prime, where the [[Characteristic (algebra)|characteristic]] of the field is not p, the [[Galois group]] of

:'''Q'''(x, y)/'''Q'''(y)