Elliptic Curve - Elliptic Curves Over The Complex Numbers

Elliptic Curves Over The Complex Numbers

The formulation of elliptic curves as the embedding of a torus in the complex projective plane follows naturally from a curious property of Weierstrass's elliptic functions. These functions and their first derivative are related by the formula

Here, g₂ and g₃ are constants; is the Weierstrass elliptic function and its derivative. It should be clear that this relation is in the form of an elliptic curve (over the complex numbers). The Weierstrass functions are doubly periodic; that is, they are periodic with respect to a lattice Λ; in essence, the Weierstrass functions are naturally defined on a torus T = C/Λ. This torus may be embedded in the complex projective plane by means of the map

This map is a group isomorphism, carrying the natural group structure of the torus into the projective plane. It is also an isomorphism of Riemann surfaces, and so topologically, a given elliptic curve looks like a torus. If the lattice Λ is related to a lattice cΛ by multiplication by a non-zero complex number c, then the corresponding curves are isomorphic. Isomorphism classes of elliptic curves are specified by the j-invariant.

The isomorphism classes can be understood in a simpler way as well. The constants g₂ and g₃, called the modular invariants, are uniquely determined by the lattice, that is, by the structure of the torus. However, the complex numbers form the splitting field for polynomials with real coefficients, and so the elliptic curve may be written as

One finds that

and

so that the modular discriminant is

Here, λ is sometimes called the modular lambda function.

Note that the uniformization theorem implies that every compact Riemann surface of genus one can be represented as a torus.

Over the complex numbers, every elliptic curve has nine inflection points. Every line through two of these points also passes through a third inflection point; the nine points and 12 lines formed in this way form a realization of the Hesse configuration.