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(n
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
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
editThe 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
Parametrization of the modular curve
editFor 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 [1] by rational functions. The simplest nontrivial example is X0(2), where:
is (up to the constant term) the McKay–Thompson series for the class 2B of the Monster, and
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
editA curve C, over Q is called a modular curve if for some n there exists a surjective morphism
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
Galois theory of the modular curve
editThe 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
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.
See also
editReferences
edit- 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
edit- OEIS sequence A001617 (Genus of modular group Gamma_0(n). Or, genus of modular curve X_0(n))
- [2] Coefficients of X0(n)