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Concept
Classical modular curve
Summary
In number theory, the classical modular curve is an irreducible plane algebraic curve given by an equation
Φn(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 Φn(x, x).
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.
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 Φn 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 Φn(x, y) = Φn(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.
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:
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
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.
Official source
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