Hyperelliptic curve
In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus g > 1, given by an equation of the form
A hyperelliptic function is an element of the function field of such a curve, or of the Jacobian variety on the curve; these two concepts are identical for elliptic functions, but different for hyperelliptic functions.
Genus
The degree of the polynomial determines the genus of the curve: 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. Meanwhile, a curve of degree 2g + 2 is termed a real hyperelliptic curve. This statement about genus remains true for g = 0 or 1, but those special cases are not called "hyperelliptic". In the case g = 1 (if one chooses a distinguished point), such a curve is called an elliptic curve.
Formulation and choice of model
While this model is the simplest way to describe hyperelliptic curves, such an equation will have a
To be more precise, the equation defines a
The glueing maps between the two charts are given by
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 = 2g + 1 and 2g + 2 can be unified, since we might as well use an automorphism of the projective plane to move any ramification point away from infinity.
Using Riemann–Hurwitz formula
Using the Riemann–Hurwitz formula, the hyperelliptic curve with genus g is defined by an equation with degree n = 2g + 2. Suppose f : X → P1 is a branched covering with ramification degree 2, where X is a curve with genus g and P1 is the Riemann sphere. Let g1 = g and g0 be the genus of P1 ( = 0 ), then the Riemann-Hurwitz formula turns out to be
where s is over all ramified points on X. The number of ramified points is n, and at each ramified point s we have es = 2, so the formula becomes
so n = 2g + 2.
Occurrence and applications
All curves of genus 2 are hyperelliptic, but for genus ≥ 3 the generic curve is not hyperelliptic. This is seen heuristically by a
The definition by quadratic extensions of the rational function field works for fields in general except in characteristic 2; in all cases the geometric definition as a ramified double cover of the projective line is available, if the extension is assumed to be separable.
Hyperelliptic curves can be used in
Hyperelliptic curves also appear composing entire connected components of certain strata of the moduli space of Abelian differentials.[2]
Hyperellipticity of genus-2 curves was used to prove
Classification
Hyperelliptic curves of given genus g have a moduli space, closely related to the ring of
History
Hyperelliptic functions were first published[
See also
References
- "Hyper-elliptic curve", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- A user's guide to the local arithmetic of hyperelliptic curves