Formal Definition
Hyperelliptic curves can be defined over fields of any characteristic. Hence we consider an arbitrary field and its algebraic closure . An (imaginary) hyperelliptic curve of genus over is given by an equation of the form
where is a polynomial of degree not larger than and is a monic polynomial of degree . Furthermore we require the curve to have no singular points. In our setting, this entails that no point satisfies both and the equations and . This definition differs from the definition of a general hyperelliptic curve in the fact that can also have degree in the general case. From now on we drop the adjective imaginary and simply talk about hyperelliptic curves, as is often done in literature. Note that the case corresponds to being a cubic polynomial, agreeing with the definition of an elliptic curve. If we view the curve as lying in the projective plane with coordinates, we see that there is a particular point lying on the curve, namely the point at infinity denoted by . So we could write .
Suppose the point not equal to lies on the curve and consider . As can be simplified to, we see that is also a point on the curve. is called the opposite of and is called a Weierstrass point if, i.e. . Furthermore, the opposite of is simply defined as .
Read more about this topic: Imaginary Hyperelliptic Curve
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