Hyperelliptic curve

A hyperelliptic curve with the equation y = sqrt(x⁴ − x² + 1). The red curve [f(x)] shows the principal square root, whereas the green curve [g(x)] shows the other square root.

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

y² = f(x)

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 f(x) 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. When the degree is equal to 2g+1, the curve is called an imaginary hyperelliptic curve.

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. It turns out that after doing this, we can cover the curve with two affine pieces: the one already given by y² = f(x) and one given by w² = v^{2g + 2}f(1 / v). The glueing maps between the two pieces are given by and , wherever they are defined.

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 the generic curve is 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.

Hyperelliptic functions were first published by Adolph Göpel (1812-1847) in his last paper Abelsche Transcendenten erster Ordnung (Abelian transcendents of first order) (in Journal für reine und angewandte Mathematik, vol. 35, 1847). Independently Johann G. Rosenhain worked on that matter and published Umkehrungen ultraelliptischer Integrale erster Gattung (in Mémoires des sa vanta etc., vol. 11, 1851).