Imaginary hyperelliptic curve

A hyperelliptic curve is a particular kind of algebraic curve. There exist hyperelliptic curves of every genus ${\displaystyle g\geq 1}$. If the genus of a hyperelliptic curve equals 1, we simply call the curve an elliptic curve. Hence we can see hyperelliptic curves as generalizations of elliptic curves. There is a well-known group structure on the set of points lying on an elliptic curve over some field ${\displaystyle K}$, which we can describe geometrically with chords and tangents. Generalizing this group structure to the hyperelliptic case is not straightforward. We cannot define the same group law on the set of points lying on a hyperelliptic curve, instead a group structure can be defined on the so-called Jacobian of a hyperelliptic curve. The computations differ depending on the number of points at infinity. This article is about imaginary hyperelliptic curves, these are hyperelliptic curves with exactly 1 point at infinity. Real hyperelliptic curves have two points at infinity.

Hyperelliptic curves can be defined over fields of any characteristic. Hence we consider an arbitrary field ${\displaystyle K}$ and its algebraic closure ${\displaystyle {\overline {K}}}$. An (imaginary) hyperelliptic curve of genus ${\displaystyle g}$ over ${\displaystyle K}$ is given by an equation of the form

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