Classical modular curve

In number theory, the classical modular curve is a plane algebraic curve given by an equation Φ_n(x, y)=0, where for the j-invariant j(τ), x=j(n τ), y=j(τ) is a point on the curve. The curve is sometimes called X₀(n), though often that is used for the abstract 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).

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. 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 particulr 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.

See also

• Algebraic curves
• J-invariant
• Modular curve
• Modular function

References

• Serge Lang, Elliptic Functions, Addison-Wesley, 1973

• Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1972