Home > Hyperelliptic curve

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 ramificationAlgebraic number theory Algebraic topology Complex analysis In mathematics, ramification is a geometric term used for 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. It is 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 spaceIn algebraic geometry, the moduli problem is to describe the parameters on which algebraic varieties depend. The use of the term modulus here for such a parameter goes back to the same source as in modular form: a modular form in general is some kind of d dimension check.

Hyperelliptic curves can be used in hyperelliptic curve cryptographyHyperelliptic curve cryptography is similar to elliptic curve cryptography (ECC) insomuch as the algebraic geometry construct of a hyperelliptic curve with an appropriate group law provides an Abelian group on which to do arithmetic. The use of hyperellip in cryptosystemA crypto system (or cryptographic system is, essentially, the package of all processes, formulae, and instructions for encoding and decoding messages using cryptography. It will generally contain an integrated assembly of cryptographic primitives (e.s based on the discrete logarithm problem.