Classical Modular Curve - Mappings

Mappings

A curve C over the rationals Q such that there exists a surjective morphism from X₀(n) to C for some n, given by a rational map with integer coefficients

φ:X₀(n) → C,

is a modular curve. The famous modularity theorem tells us that all elliptic curves over Q are modular.

Mappings also arise in connection with X₀(n) since points on it correspond to n-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 n, the cyclic isogenies, correspond to points on X₀(n).

When X₀(n) has genus one, it will itself be isomorphic to an elliptic curve, which will have the same j-invariant. For instance, X₀(11) has j-invariant -122023936/161051 = - 21211-5313, and is isomorphic to the curve y2+y = x3-x2-10x-20. If we substitute this value of j for y in X₀(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 x and y, all on X₀(5), corresponding to the six isogenies between these three curves. If in the curve y2+y = x3-x2-10x-20 isomorphic to X₀(11) we substitute

and

and factor, we get an extraneous factor of a rational function of x, and the curve y^2+y=x^3-x^2, with j-invariant -4096/11. Hence both curves are modular of level 11, having mappings from X₀(11).

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 φ:X₀(n) → 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.