Real hyperelliptic curve
There are two types of hyperelliptic curves, a class of algebraic curves: real hyperelliptic curves and imaginary hyperelliptic curves which differ by the number of points at infinity. Hyperelliptic curves exist for every genus . The general formula of Hyperelliptic curve over a finite field  is given by 

where  satisfy certain conditions. In this page, we describe more about real hyperelliptic curves, these are curves having two points at infinity while imaginary hyperelliptic curves have one point at infinity. Definition
A real hyperelliptic curve of genus g over K is defined by an equation of the form  where  has degree not larger than g+1 while  must have degree 2g+1 or 2g+2. This curve is a non singular curve where no point  in the algebraic closure of  satisfies the curve equation  and both partial derivative equations:  and . The set of (finite) –rational points on C is given by

where  is the set of points at infinity. For real hyperelliptic curves, there are two points at infinity,  and . For any point , the opposite point of  is given by ; it is the other point with x-coordinate a that also lies on the curve. Example
Let  where

over . Since  and  has degree 6, thus  is a curve of genus g = 2. The homogeneous version of the curve equation is given by

It has a single point at infinity given by (0:1:0) but this point is singular. The blowup of  has 2 different points at infinity, which we denote  and . Hence this curve is an example of a real hyperelliptic curve. In general, every curve given by an equation where f has even degree has two points at infinity and is a real hyperelliptic curve while those where f has odd degree have only a single point in the blowup over (0:1:0) and are thus imaginary hyperelliptic curves. In both cases this assumes that the affine part of the curve is non-singular (see the conditions on the derivatives above)

Arithmetic in a real hyperelliptic curve

In real hyperelliptic curve, addition is no longer defined on points as in elliptic curves but on divisors and the Jacobian. Let  be a hyperelliptic curve of genus g over a finite field K. A divisor  on  is a formal finite sum of points  on . We write
 where  and  for almost all . The degree of  is defined by 

 is said to be defined over  if  for all automorphisms σ of  over . The set  of divisors of  defined over  forms an additive abelian group under the addition rule 

The set  of all degree zero divisors of  defined over  is a subgroup of . We take an example:

Let  and . If we add them then . The degree of  is  and the degree of  is . Then, 

For polynomials , the divisor of  is defined by

If the function  has a pole at a point  then  is the order of vanishing of  at . Assume  are polynomials in ; the divisor of the rational function  is called a principal divisor and is defined by . We denote the group of principal divisors by , i.e., . The Jacobian of  over  is defined by . The factor group  is also called the divisor class group of . The elements which are defined over  form the group . We denote by  the class of  in . There are two canonical ways of representing divisor classes for real hyperelliptic curves  which have two points infinity . The first one is to represent a degree zero divisor by  such that , where ,, and  if  The representative  of  is then called semi reduced. If  satisfies the additional condition  then the representative  is called reduced. Notice that  is allowed for some i. It follows that every degree 0 divisor class contain a unique representative  with 

where  is divisor that is coprime with both  and , and . The other representation is balanced at infinity. Let , note that this divisor is -rational even if the points  and  are not independently so. Write the representative of the class  as ,
where  is called the affine part and does not contain  and , and let . If  is even then

If  is odd then 

For example, let the affine parts of two divisors be given by
 and 
then the balanced divisors are
 and

Transformation from real hyperelliptic curve to imaginary hyperelliptic curve

Let  be a real quadratic curve over a field . If there exists a ramified prime divisor of degree 1 in  then we are able to perform a birational transformation to an imaginary quadratic curve. A (finite or infinite) point is said to be ramified if it is equal to its own opposite. It means that , i.e. that . If  is ramified then  is a ramified prime divisor. The real hyperelliptic curve  of genus  with a ramified -rational finite point  is birationally equivalent to an imaginary model  of genus , i.e. and the function fields are equal . Here:

In our example  where , h(x) is equal to 0. For any point ,  is equal to 0 and so the requirement for P to be ramified becomes . Substituting  and , we obtain , where , i.e., . From (), we obtain  and . For g = 2, we have . For example, let  then  and , we obtain 

To remove the denominators this expression is multiplied by , then:

giving the curve 
 where 

 is an imaginary quadratic curve since  has degree . References

Algebraic curves