Classical modular curve

Mappings also arise in connection with X0(n) since points on it correspond to some n-isogenous pairs of elliptic curves.

An isogeny between two elliptic curves is a non-trivial morphism of varieties (defined by a rational map) between the curves which also respects the group laws, and hence which sends the point at infinity (serving as the identity of the group law) to the point at infinity.

Such a map is always surjective and has a finite kernel, the order of which is the degree of the isogeny.

Points on X0(n) correspond to pairs of elliptic curves admitting an isogeny of degree n with cyclic kernel.

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.

The Galois theory of the modular curve was investigated by Erich Hecke.

In the case of X0(p) with p prime, where the characteristic of the field is not p, 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 p elements, which has p + 1 points, the degree of X0(p).

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