Classical modular curve
- Φn(x, y) = 0,
such that (x, y) = (j(nτ), j(τ)) is a point on the curve. Here j(τ) denotes the j-invariant.
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 Φn(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 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 Φn 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 ψ(n), where ψ is the Dedekind psi function. Since Φn(x, y) = Φn(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
parametrizes X0(2) in terms of rational functions of j2. It is not necessary to actually compute j2 to use this parametrization; it can be taken as an arbitrary parameter.
A curve C, over Q is called a modular curve if for some n there exists a surjective morphism φ : X0(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 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.
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 −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 factor, we get an extraneous factor of a rational function of x, and the curve y2 + y = x3 − x2, with j-invariant −21211−1. 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.
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 Φ0(n) is a polynomial of degree 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 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).
This extension contains an algebraic extension F/Q where if in the notation of Gauss then:
If we extend the field of constants to be F, we now have an extension with Galois group PSL(2, p), the projective special linear group of the field with p elements, which is a finite simple group. By specializing y 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 F, and PGL(2, p) over Q.
When n is not a prime, the Galois groups can be analyzed in terms of the factors of n as a wreath product.
- 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 
- 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