Classical modular curve

Template:Short description In number theory, the classical modular curve is an irreducible plane algebraic curve given by an equation

Template:Math,

such that Template:Math is a point on the curve. Here Template:Math denotes the [[j-invariant|Template:Mvar-invariant]].

The curve is sometimes called Template:Math, 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 Template:Math.

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 Template:Math.

Geometry of the modular curve

File:Modknot11.png

Knot at infinity of Template:Math

The classical modular curve, which we will call Template:Math, is of degree greater than or equal to Template:Math when Template:Math, with equality if and only if Template:Mvar is a prime. The polynomial Template:Math 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 Template:Mvar with coefficients in Template:Math, it has degree Template:Math, where Template:Mvar is the Dedekind psi function. Since Template:Math, Template:Math is symmetrical around the line Template:Math, 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 Template:Math, there are two singularities at infinity, where Template:Math and Template:Math, 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 Template:Math, or Template:Math, Template:Math has genus zero, and hence can be parametrized [1] by rational functions. The simplest nontrivial example is Template:Math, where:

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

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

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

$y=\frac{(j_2+16{)}^3}{j_2}$

parametrizes Template:Math in terms of rational functions of Template:Math. It is not necessary to actually compute Template:Math to use this parametrization; it can be taken as an arbitrary parameter.

Mappings

A curve Template:Mvar, over Template:Math is called a modular curve if for some Template:Mvar there exists a surjective morphism Template:Math, given by a rational map with integer coefficients. The famous modularity theorem tells us that all elliptic curves over Template:Math are modular.

Mappings also arise in connection with Template:Math since points on it correspond to some Template:Mvar-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 Template:Math correspond to pairs of elliptic curves admitting an isogeny of degree Template:Mvar with cyclic kernel.

When Template:Math has genus one, it will itself be isomorphic to an elliptic curve, which will have the same [[j-invariant|Template:Mvar-invariant]].

For instance, Template:Math has Template:Mvar-invariant Template:Math, and is isomorphic to the curve Template:Math. If we substitute this value of Template:Mvar for Template:Mvar in Template:Math, 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 Template:Mvar and Template:Mvar, all on Template:Math, corresponding to the six isogenies between these three curves.

If in the curve Template:Math, isomorphic to Template:Math we substitute

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

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

and factor, we get an extraneous factor of a rational function of Template:Mvar, and the curve Template:Math, with Template:Mvar-invariant Template:Math. Hence both curves are modular of level Template:Math, having mappings from Template:Math.

By a theorem of Henri Carayol, if an elliptic curve Template:Mvar is modular then its conductor, an isogeny invariant described originally in terms of cohomology, is the smallest integer Template:Mvar such that there exists a rational mapping Template:Math. Since we now know all elliptic curves over Template:Math are modular, we also know that the conductor is simply the level Template:Mvar 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 Template:Math, the modular equation Template:Math is a polynomial of degree Template:Math in Template:Mvar, whose roots generate a Galois extension of Template:Math. In the case of Template:Math with Template:Mvar prime, where the characteristic of the field is not Template:Mvar, the Galois group of Template:Math is Template:Math, the projective general linear group of linear fractional transformations of the projective line of the field of Template:Mvar elements, which has Template:Math points, the degree of Template:Math.

This extension contains an algebraic extension Template:Math where if $p^{*}=(-1{)}^{(p-1)/2}p$ in the notation of Gauss then:

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

If we extend the field of constants to be Template:Mvar, we now have an extension with Galois group Template:Math, the projective special linear group of the field with Template:Mvar elements, which is a finite simple group. By specializing Template:Mvar to a specific field element, we can, outside of a thin set, obtain an infinity of examples of fields with Galois group Template:Math over Template:Mvar, and Template:Math over Template:Math.

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

See also

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

References

• , 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

• Template:OEIS el
• [2] Coefficients of Template:Math