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,

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 notation 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).

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

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 singularities 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

For 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:

${\displaystyle j_{2}(q)=q^{-1}-24+276q-2048q^{2}+11202q^{3}+\cdots =\left({\frac {\eta (q)}{\eta (q^{2})}}\right)^{24}}$

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

${\displaystyle x={\frac {(j_{2}+256)^{3}}{j_{2}^{2}}},}$

${\displaystyle y={\frac {(j_{2}+16)^{3}}{j_{2}}}}$

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 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

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

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

and factor, we get an extraneous factor of a rational function of x, and the curve 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 ${\displaystyle p^{*}=(-1)^{(p-1)/2}p}$ in the notation of Gauss then:

${\displaystyle F=\mathbf {Q} \left({\sqrt {p^{*}}}\right).}$

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.

See also

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

References

• Hecke, Erich (1935), "Die eindeutige Bestimmung der Modulfunktionen q-ter Stufe durch algebraische Eigenschaften", Mathematische Annalen, 111: 293–301, doi:10.1007/BF01472221, 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

External links

• OEIS sequence A001617 (Genus of modular group Gamma_0(n). Or, genus of modular curve X_0(n))
• Coefficients of X₀(n)