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,

where for the j-invariant j(τ),

x=j(n τ), y=j(τ)

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 Φ_n(x, x).

Geometry of the modular curve

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 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 x with coefficients in Z[y], it has degree ψ(n), where ψ is the Dedekind psi function. Since Φ_n(x, y) =   Φ_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

When n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, or 25, X₀(n) has genus zero, and hence can be parametrized by rational functions. The simplest nontrivial example is X₀(2), where if

$j_2(q) = q^{-1} - 24 + 276q -2048q^2 + 11202q^3 + \cdots = ((\eta(q)/\eta(q^2))^{24}$

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

$x = \frac{(j_2+256)^3}{j_2^2}, y = \frac{(j_2+16)^3}{j_2}$

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.

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

$x \mapsto \frac{x^5-2x^4+3x^3-2x+1}{x^2(x-1)^2}$

and

$y \mapsto y-\frac{(2y+1)(x^4+x^3-3x^2+3x-1)}{x^3(x-1)^3}$

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.

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 PGL₂(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

$F = \mathbf {Q}\left(\sqrt{(-1)^\frac{p-1}{2}p}\right)$

of Q. If we extend the field of constants to be F, we now have an extension with Galois group PSL₂(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 PSL₂(p) over F, and PGL₂(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.

See also

• Algebraic curves
• J-invariant
• Modular curve
• Modular function

External links

• Sequence  A001617 in the OEIS: Genus of X₀(n)
• [1] Coefficients of X₀(n)

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[2]
• 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