Hyperelliptic curve

This statement about genus remains true for g = 0 or 1, but those special cases are not called "hyperelliptic".

While this model is the simplest way to describe hyperelliptic curves, such an equation will have a singular point at infinity in the projective plane.

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 = 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 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.

Counting constants, with n = 2g + 2, the collection of n points subject to the action of the automorphisms of the projective line has (2g + 2) − 3 degrees of freedom, which is less than 3g − 3, the number of moduli of a curve of genus g, unless g is 2.

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 also appear composing entire connected components of certain strata of the moduli space of Abelian differentials.

Hyperelliptic curves of given genus g have a moduli space, closely related to the ring of invariants of a binary form of degree 2g+2.

[specify] Hyperelliptic functions were first published[citation needed] by Adolph Göpel (1812-1847) in his last paper Abelsche Transcendenten erster Ordnung (Abelian transcendents of first order) (in Journal für die reine und angewandte Mathematik, vol.

Independently Johann G. Rosenhain worked on that matter and published Umkehrungen ultraelliptischer Integrale erster Gattung (in Mémoires des savants etc., vol.