Hyperelliptic function

In algebraic geometry, a hyperelliptic curve (over the complex numbers) is an algebraic curve given by an equation of the form

<math>y^2 = f(x)<math>

where f(x) is a polynomial of degree n > 4 with n distinct roots. A hyperelliptic function is a function from the function field of such a curve; or possibly on the Jacobian variety on the curve, these being two concepts that are the same for the elliptic function case, but different in this case.

If n is a cubic or quartic polynomial, then the resulting curve is an elliptic curve.

The genus of a hyperelliptic curve determines the degree: a polynomial of degree 2g+1 or 2g+2 gives a curve of genus g.

While this model is the simplest way to describe hyperelliptic curves, it should be noted that such an equation will have a singular point at infinity in the projective plane, a feature specific to the case n > 4. Therefore in giving such an equation to specify a non-singular curve, it is almost always assumed that a non-singular model, equivalent in the sense of birational geometry, is meant. To be more precise, the equation defines a quadratic extension of C(x), and it is that function field that is meant. The singular point at infinity can be removed (since this is a curve) by the normalization (integral closure) process.

In fact geometric shorthand is assumed, with the curve C being defined as a ramified double cover of the projective line, the ramification occurring at the roots of f, and also for odd n at the point at infinity. In this way the cases n = 2m − 1 and 2m can be unified, since we might as well use an automorphism of the projective line to move any ramification point away from infinity.

All curves of genus 2 are hyperelliptic, but for genus ≥ 3 there are curves that are not hyperelliptic. This is shown by a moduli space dimension check.

Hyperelliptic curves can be used in hyperelliptic curve cryptography in cryptosystems based on the discrete logarithm problem.