Classical modular curve

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

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 $X 0 (n)$ , though often that 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 $Φ n (x, x)$ .

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 $X 0 (n)$ , is of degree greater than or equal to $2 n$ 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 $Φ n (x, y) = Φ n (y, x)$ , $X 0 (n)$ 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 singularites at infinity, where $x = 0, y = \infty$ and $x = \infty, y = 0$ , which have only one branch and hence have a knot invariant which is a true knot, and not just a link.

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