Classical modular curve: Difference between revisions

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== Geometry of the modular curve ==

[[Image:Modknot11.png|thumb|Knot at infinity of X₀(11)]]

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 Φ_n has integer coefficients, and hence is defined over every field. However, butthe thesecoefficients are sufficiently large, makingthat computational work with the curve computationallycan 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), 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 singularites 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.

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.

Φ_n has integer coefficients, and hence is defined over every field, but these are large, making the curve computationally 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), 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 singularites 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.

== Parametrization of the modular curve ==