Real hyperelliptic curve

A hyperelliptic curve is a class of algebraic curves. Hyperelliptic curves exist for every genus formula_1. The general formula of Hyperelliptic curve over a finite field formula_2 is given by 
where formula_4 satisfy certain conditions. There are two types of hyperelliptic curves: real hyperelliptic curves and imaginary hyperelliptic curves which differ by the number of points at infinity. 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 formula_5 where formula_6 has degree not larger than "g+1" while formula_7 must have degree "2g+1" or "2g+2". This curve is a non singular curve where no point formula_8 in the algebraic closure of formula_2 satisfies the curve equation formula_10 and both partial derivative equations: formula_11 and formula_12.
The set of (finite) formula_13–rational points on "C" is given by
Where formula_15 is the set of points at infinity. For real hyperelliptic curves, there are two points at infinity, formula_16 and formula_17. For any point formula_18, the opposite point of formula_19 is given by formula_20; it is the other point with "x"-coordinate "a" that also lies on the curve.
Example.
Let formula_21 where
over formula_23. Since formula_24 and formula_25 has degree 6, thus formula_26 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 formula_26 has 2 different points at infinity, which we denote formula_29and formula_30. 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 nonsingular (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 formula_26 be a hyperelliptic curve of genus "g" over a finite field "K". A divisor formula_32 on formula_26 is a formal finite sum of points formula_19 on formula_26. We write 
The degree of formula_40 is defined by 
formula_32 is said to be defined over formula_43 if formula_44 for all automorphisms σ of formula_45 over formula_2 . The set formula_47 of divisors of formula_26 defined over formula_2 forms an additive abelian group under the addition rule 
The set formula_51 of all degree zero divisors of formula_26 defined over formula_2 is a subgroup of formula_47.
We take an example:
Let formula_55 and formula_56. If we add them then formula_57. The degree of formula_58 is formula_59 and the degree of formula_60 is formula_61. 
Then, formula_62
For polynomials formula_63, the divisor of formula_64 is defined by
formula_64 has a pole at a point formula_19 then formula_68 is the order of vanishing of formula_64 at formula_19. Assume formula_71 are polynomials in formula_72; the divisor of the rational function formula_73 is called a principal divisor and is defined by formula_74. We denote the group of principal divisors by formula_75, i.e. formula_76. The Jacobian of formula_26 over formula_2 is defined by formula_79. The factor group formula_80 is also called the divisor class group of formula_26. The elements which are defined over formula_2 form the group formula_83. We denote by formula_84 the class of formula_32 in formula_86.
There are two canonical ways of representing divisor classes for real hyperelliptic curves formula_26 which have two points infinity formula_88. The first one is to represent a degree zero divisor by formula_89 such that formula_90, where formula_91,formula_92, and formula_93 if formula_94 The representative formula_32 of formula_96 is then called semi reduced. If formula_32 satisfies the additional condition formula_98 then the representative formula_32 is called reduced. Notice that formula_100 is allowed for some i. It follows that every degree 0 divisor class contain a unique representative formula_96 with 
where formula_103 is divisor that is coprime with both 
The other representation is balanced at infinity.
Let formula_107, note that this divisor is formula_2-rational even if the points formula_29 and formula_30 are not independently so. Write the representative of the class formula_96 as formula_112,
where formula_58 is called the affine part and does not contain formula_16 and formula_17, and let formula_116. If formula_117 is even then
If formula_117 is odd then 
For example, let the affine parts of two divisors be given by
then the balanced divisors are
Transformation from real hyperelliptic curve to imaginary hyperelliptic curve.
Let formula_26 be a real quadratic curve over a field formula_2. If there exists a ramified prime divisor of degree 1 in formula_2 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 formula_128, i.e. that formula_129. If formula_19 is ramified then formula_131 is a ramified prime divisor.
The real hyperelliptic curve formula_5 of genus formula_133 with a ramified formula_2-rational finite point formula_135 is birationally equivalent to an imaginary model formula_136 of genus formula_133, i.e. formula_138 and the function fields are equal formula_139. Here:
In our example formula_21 where formula_143, "h(x)" is equal to 0. For any point formula_135, formula_145 is equal to 0 and so the requirement for "P" to be ramified becomes formula_146. Substituting formula_145 and formula_148, we obtain formula_149, where formula_150, i.e. formula_151.
From (i), we obtain formula_152 and formula_153 . For g=2, we have formula_154
For example, let formula_155 then formula_156 and formula_157, we obtain 
To remove the denominators this expression is multiplied by formula_159, then:
giving the curve 
formula_163 is an imaginary quadratic curve since formula_164 has degree formula_165.