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 ${\displaystyle C}$ be a hyperelliptic curve of genus g over a finite field K. A divisor ${\displaystyle D}$ on ${\displaystyle C}$ is a formal finite sum of points ${\displaystyle P}$ on ${\displaystyle C}$. We write $${\displaystyle D=\sum _{P\in C}{n_{P}P}}$$ where ${\displaystyle n_{P}\in \mathbb {Z} }$ and ${\displaystyle n_{p}=0}$ for almost all ${\displaystyle P}$.

The degree of ${\textstyle D=\sum _{P\in C}{n_{P}P}}$ is defined by $${\displaystyle \deg(D)=\sum _{P\in C}{n_{P}}.}$$ ${\displaystyle D}$ is said to be defined over ${\displaystyle K}$ if ${\textstyle D^{\sigma }=\sum _{P\in C}n_{P}P^{\sigma }=D}$ for all automorphisms σ of ${\displaystyle {\overline {K}}}$ over ${\displaystyle K}$. The set ${\displaystyle Div(K)}$ of divisors of ${\displaystyle C}$ defined over ${\displaystyle K}$ forms an additive abelian group under the addition rule $${\displaystyle \sum a_{P}P+\sum b_{P}P=\sum {(a_{P}+b_{P})P}.}$$

The set ${\displaystyle Div^{0}(K)}$ of all degree zero divisors of ${\displaystyle C}$ defined over ${\displaystyle K}$ is a subgroup of ${\displaystyle Div(K)}$.

We take an example:

Let ${\displaystyle D_{1}=6P_{1}+4P_{2}}$ and ${\displaystyle D_{2}=1P_{1}+5P_{2}}$. If we add them then ${\displaystyle D_{1}+D_{2}=7P_{1}+9P_{2}}$. The degree of ${\displaystyle D_{1}}$ is ${\displaystyle \deg(D_{1})=6+4=10}$ and the degree of ${\displaystyle D_{2}}$ is ${\displaystyle \deg(D_{2})=1+5=6}$. Then, ${\displaystyle \deg(D_{1}+D_{2})=\deg(D_{1})+\deg(D_{2})=16.}$

For polynomials ${\displaystyle G\in K[C]}$, the divisor of ${\displaystyle G}$ is defined by $${\displaystyle \mathrm {div} (G)=\sum _{P\in C}{\mathrm {ord} }_{P}(G)P.}$$ If the function ${\displaystyle G}$ has a pole at a point ${\displaystyle P}$ then ${\displaystyle -{\mathrm {ord} }_{P}(G)}$ is the order of vanishing of ${\displaystyle G}$ at ${\displaystyle P}$. Assume ${\displaystyle G,H}$ are polynomials in ${\displaystyle K[C]}$; the divisor of the rational function ${\displaystyle F=G/H}$ is called a principal divisor and is defined by ${\displaystyle \mathrm {div} (F)=\mathrm {div} (G)-\mathrm {div} (H)}$. We denote the group of principal divisors by ${\displaystyle P(K)}$, i.e., ${\displaystyle P(K)=\{\mathrm {div} (F)\mid F\in K(C)\}}$. The Jacobian of ${\displaystyle C}$ over ${\displaystyle K}$ is defined by ${\displaystyle J=Div^{0}/P}$. The factor group ${\displaystyle J}$ is also called the divisor class group of ${\displaystyle C}$. The elements which are defined over ${\displaystyle K}$ form the group ${\displaystyle J(K)}$. We denote by ${\displaystyle {\overline {D}}\in J(K)}$ the class of ${\displaystyle D}$ in ${\displaystyle Div^{0}(K)/P(K)}$.

There are two canonical ways of representing divisor classes for real hyperelliptic curves ${\displaystyle C}$ which have two points infinity ${\displaystyle S=\{\infty _{1},\infty _{2}\}}$. The first one is to represent a degree zero divisor by ${\displaystyle {\bar {D}}}$ such that ${\textstyle D=\sum _{i=1}^{r}P_{i}-r\infty _{2}}$, where ${\displaystyle P_{i}\in C({\bar {\mathbb {F} }}_{q})}$,${\displaystyle P_{i}\not =\infty _{2}}$, and ${\displaystyle P_{i}\not ={\bar {P_{j}}}}$ if ${\displaystyle i\neq j}$ The representative ${\displaystyle D}$ of ${\displaystyle {\bar {D}}}$ is then called semi reduced. If ${\displaystyle D}$ satisfies the additional condition ${\displaystyle r\leq g}$ then the representative ${\displaystyle D}$ is called reduced.^[1] Notice that ${\displaystyle P_{i}=\infty _{1}}$ is allowed for some i. It follows that every degree 0 divisor class contain a unique representative ${\displaystyle {\bar {D}}}$ with $${\displaystyle D=D_{x}-\deg(D_{x})\infty _{2}+v_{1}(D)(\infty _{1}-\infty _{2}),}$$ where ${\displaystyle D_{x}}$ is divisor that is coprime with both ${\displaystyle \infty _{1}}$ and ${\displaystyle \infty _{2}}$, and ${\displaystyle 0\leq \deg(D_{x})+v_{1}(D)\leq g}$.

The other representation is balanced at infinity. Let ${\displaystyle D_{\infty }=\infty _{1}+\infty _{2}}$, note that this divisor is ${\displaystyle K}$-rational even if the points ${\displaystyle \infty _{1}}$ and ${\displaystyle \infty _{2}}$ are not independently so. Write the representative of the class ${\displaystyle {\bar {D}}}$ as ${\displaystyle D=D_{1}+D_{\infty }}$, where ${\displaystyle D_{1}}$ is called the affine part and does not contain ${\displaystyle \infty _{1}}$ and ${\displaystyle \infty _{2}}$, and let ${\displaystyle d=\deg(D_{1})}$. If ${\displaystyle d}$ is even then $${\displaystyle D_{\infty }={\frac {d}{2}}(\infty _{1}+\infty _{2}).}$$

If ${\displaystyle d}$ is odd then $${\displaystyle D_{\infty }={\frac {d+1}{2}}\infty _{1}+{\frac {d-1}{2}}\infty _{2}.}$$ For example, let the affine parts of two divisors be given by

${\displaystyle D_{1}=6P_{1}+4P_{2}}$ and ${\displaystyle D_{2}=1P_{1}+5P_{2}}$

then the balanced divisors are

${\displaystyle D_{1}=6P_{1}+4P_{2}-5D_{\infty _{1}}-5D_{\infty _{2}}}$ and ${\displaystyle D_{2}=1P_{1}+5P_{2}-3D_{\infty _{1}}-3D_{\infty _{2}}}$

Let ${\displaystyle C}$ be a real quadratic curve over a field ${\displaystyle K}$. If there exists a ramified prime divisor of degree 1 in ${\displaystyle K}$ 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 ${\displaystyle P=(a,b)={\overline {P}}=(a,-b-h(a))}$, i.e. that ${\displaystyle h(a)+2b=0}$. If ${\displaystyle P}$ is ramified then ${\displaystyle D=P-\infty _{1}}$ is a ramified prime divisor.^[2]

The real hyperelliptic curve ${\displaystyle C:y^{2}+h(x)y=f(x)}$ of genus ${\displaystyle g}$ with a ramified ${\displaystyle K}$-rational finite point ${\displaystyle P=(a,b)}$ is birationally equivalent to an imaginary model ${\displaystyle C':y'^{2}+{\bar {h}}(x')y'={\bar {f}}(x')}$ of genus ${\displaystyle g}$, i.e. ${\displaystyle \deg({\bar {f}})=2g+1}$ and the function fields are equal ${\displaystyle K(C)=K(C')}$.^[3] Here:

${\displaystyle x'={\frac {1}{x-a}}}$ and ${\displaystyle y'={\frac {y+b}{(x-a)^{g+1}}}}$

i

In our example ${\displaystyle C:y^{2}=f(x)}$ where ${\displaystyle f(x)=x^{6}+3x^{5}-5x^{4}-15x^{3}+4x^{2}+12x}$, h(x) is equal to 0. For any point ${\displaystyle P=(a,b)}$, ${\displaystyle h(a)}$ is equal to 0 and so the requirement for P to be ramified becomes ${\displaystyle b=0}$. Substituting ${\displaystyle h(a)}$ and ${\displaystyle b}$, we obtain ${\displaystyle f(a)=0}$, where ${\displaystyle f(a)=a(a-1)(a-2)(a+1)(a+2)(a+3)}$, i.e., ${\displaystyle a\in \{0,1,2,-1,-2,-3\}}$.

From (i), we obtain ${\textstyle x={\frac {ax'+1}{x'}}}$ and ${\textstyle y={\frac {y'}{x'^{g+1}}}}$. For g = 2, we have ${\textstyle y={\frac {y'}{x'^{3}}}}$.

For example, let ${\displaystyle a=1}$ then ${\textstyle x={\frac {x'+1}{x'}}}$ and ${\textstyle y={\frac {y'}{x'^{3}}}}$, we obtain $${\displaystyle \left({\frac {y'}{x'^{3}}}\right)^{2}={\frac {x'+1}{x'}}\left({\frac {x'+1}{x'}}+1\right)\left({\frac {x'+1}{x'}}+2\right)\left({\frac {x'+1}{x'}}+3\right)\left({\frac {x'+1}{x'}}-1\right)\left({\frac {x'+1}{x'}}-2\right).}$$

To remove the denominators this expression is multiplied by ${\displaystyle x^{6}}$, then: $${\displaystyle y'^{2}=(x'+1)(2x'+1)(3x'+1)(4x'+1)(1)(1-x')}$$ giving the curve $${\displaystyle C':y'^{2}={\bar {f}}(x')}$$ where $${\displaystyle {\bar {f}}(x')=(x'+1)(2x'+1)(3x'+1)(4x'+1)(1)(1-x')=-24x'^{5}-26x'^{4}+15x'^{3}+25x'^{2}+9x'+1.}$$

${\displaystyle C'}$ is an imaginary quadratic curve since ${\displaystyle {\bar {f}}(x')}$ has degree ${\displaystyle 2g+1}$.