Английская Википедия:Classical modular curve
In number theory, the classical modular curve is an irreducible plane algebraic curve given by an equation
such that Шаблон:Math is a point on the curve. Here Шаблон:Math denotes the [[j-invariant|Шаблон:Mvar-invariant]].
The curve is sometimes called Шаблон: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 Шаблон:Math.
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 Шаблон:Math.
Geometry of the modular curve
The classical modular curve, which we will call Шаблон:Math, is of degree greater than or equal to Шаблон:Math when Шаблон:Math, with equality if and only if Шаблон:Mvar is a prime. The polynomial Шаблон: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 Шаблон:Mvar with coefficients in Шаблон:Math, it has degree Шаблон:Math, where Шаблон:Mvar is the Dedekind psi function. Since Шаблон:Math, Шаблон:Math is symmetrical around the line Шаблон: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 Шаблон:Math, there are two singularities at infinity, where Шаблон:Math and Шаблон: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 Шаблон:Math, or Шаблон:Math, Шаблон:Math has genus zero, and hence can be parametrized [1] by rational functions. The simplest nontrivial example is Шаблон:Math, where:
- <math>j_2(q)= q^{-1} - 24 + 276q -2048q^2 + 11202q^3 + \cdots =\left (\frac{\eta(q)}{\eta(q^2)} \right)^{24}</math>
is (up to the constant term) the McKay–Thompson series for the class 2B of the Monster, and Шаблон:Mvar is the Dedekind eta function, then
- <math>x = \frac{(j_2+256)^3}{j_2^2},</math>
- <math>y = \frac{(j_2+16)^3}{j_2}</math>
parametrizes Шаблон:Math in terms of rational functions of Шаблон:Math. It is not necessary to actually compute Шаблон:Math to use this parametrization; it can be taken as an arbitrary parameter.
Mappings
A curve Шаблон:Mvar, over Шаблон:Math is called a modular curve if for some Шаблон:Mvar there exists a surjective morphism Шаблон:Math, given by a rational map with integer coefficients. The famous modularity theorem tells us that all elliptic curves over Шаблон:Math are modular.
Mappings also arise in connection with Шаблон:Math since points on it correspond to some Шаблон: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 Шаблон:Math correspond to pairs of elliptic curves admitting an isogeny of degree Шаблон:Mvar with cyclic kernel.
When Шаблон:Math has genus one, it will itself be isomorphic to an elliptic curve, which will have the same [[j-invariant|Шаблон:Mvar-invariant]].
For instance, Шаблон:Math has Шаблон:Mvar-invariant Шаблон:Math, and is isomorphic to the curve Шаблон:Math. If we substitute this value of Шаблон:Mvar for Шаблон:Mvar in Шаблон: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 Шаблон:Mvar and Шаблон:Mvar, all on Шаблон:Math, corresponding to the six isogenies between these three curves.
If in the curve Шаблон:Math, isomorphic to Шаблон:Math we substitute
- <math>x \mapsto \frac{x^5-2x^4+3x^3-2x+1}{x^2(x-1)^2}</math>
- <math>y \mapsto y-\frac{(2y+1)(x^4+x^3-3x^2+3x-1)}{x^3(x-1)^3}</math>
and factor, we get an extraneous factor of a rational function of Шаблон:Mvar, and the curve Шаблон:Math, with Шаблон:Mvar-invariant Шаблон:Math. Hence both curves are modular of level Шаблон:Math, having mappings from Шаблон:Math.
By a theorem of Henri Carayol, if an elliptic curve Шаблон:Mvar is modular then its conductor, an isogeny invariant described originally in terms of cohomology, is the smallest integer Шаблон:Mvar such that there exists a rational mapping Шаблон:Math. Since we now know all elliptic curves over Шаблон:Math are modular, we also know that the conductor is simply the level Шаблон: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 Шаблон:Math, the modular equation Шаблон:Math is a polynomial of degree Шаблон:Math in Шаблон:Mvar, whose roots generate a Galois extension of Шаблон:Math. In the case of Шаблон:Math with Шаблон:Mvar prime, where the characteristic of the field is not Шаблон:Mvar, the Galois group of Шаблон:Math is Шаблон:Math, the projective general linear group of linear fractional transformations of the projective line of the field of Шаблон:Mvar elements, which has Шаблон:Math points, the degree of Шаблон:Math.
This extension contains an algebraic extension Шаблон:Math where if <math>p^* = (-1)^{(p-1)/2}p</math> in the notation of Gauss then:
- <math>F = \mathbf{Q}\left(\sqrt{p^*}\right).</math>
If we extend the field of constants to be Шаблон:Mvar, we now have an extension with Galois group Шаблон:Math, the projective special linear group of the field with Шаблон:Mvar elements, which is a finite simple group. By specializing Шаблон:Mvar to a specific field element, we can, outside of a thin set, obtain an infinity of examples of fields with Galois group Шаблон:Math over Шаблон:Mvar, and Шаблон:Math over Шаблон:Math.
When Шаблон:Mvar is not a prime, the Galois groups can be analyzed in terms of the factors of Шаблон:Mvar as a wreath product.
See also
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]Шаблон:Dead link
- 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
- Шаблон:OEIS el
- [3] Coefficients of Шаблон:Math
