REAL HYPERELLIPTIC CURVE[edit]

Hyperelliptic curve is an algebraic curve for every genus $g\geq 1$ . The general formula of Hyperelliptic curve over a finite field $K$ is given by $C:y^{2}+h(x)y=f(x)\in K[x,y]$ where $h(x),f(x)\in K$ satisfy certain condition. There are two conditions of hyperelliptic curve, real hyperelliptic curve and imaginary hyperelliptic curve. In this page, we describe more about real hyperelliptic curve in which it has two infinity points while imaginary hyperelliptic curve has one infinity point.

Definition[edit]

A real hyperelliptic curve of genus $g$ over $K$ is defined by an equation of the form $C:y^{2}+h(x)y=f(x)$ where $h(x)\in K$ has degree not larger than $g+1$ while $f(x)\in K$ must have degree $2g+1$ or $2g+2$ . This curve is a non singular curve where no point $(x,y)$ in the algebraic closure of $K$ satisfies curve equation $y^{2}+h(x)y=f(x)$ and both partial derivative equations: $2y+h(x)=0$ and $h'(x)y=f'(x)$ . The set of (finite) $K$ – rational points on C is given by

C=\{(a,b)\in E^{2}|b^{2}+h(a)y=f(b)\}\cup \{S\}

Where $S$ is set of point at infinity. In real hyperelliptic curve, there are two points at infinity, $\infty _{1}$ and $\infty _{2}$ . For any point $P(a,b)\in C(K)$ , the opposite point of $P$ is called ${\overline {P}}=(a,-b-h)$ that also lies on the curve.

Example[edit]

Let $C:y^{2}=f(x)$ where $f(x)=x^{6}+3x^{5}-5x^{4}-15x^{3}+4x^{2}+12x=x(x-1)(x-2)(x+1)(x+2)(x+3)$ over $R$ . Since $degf(x)=2g+2$ and $f(x)$ has degree 6, thus $C$ is a curve of genus $g=2$ .The curve is described in the picture below.

alt text

C:y^{2}=f(x)=x^{6}+3x^{5}-5x^{4}-15x^{3}+4x^{2}+12x

Arithmetic in Real Hyperelliptic Curve[edit]

In real hyperelliptic curve, addition is no longer defined on points as in elliptic curve but on divisors and the Jacobian. Let $C$ be a hyperelliptic curve of genus $g$ over a finite field $K$ . A divisor $D$ on $C$ is a formal finite sum of points $P$ on $C$ . We write $D=\sum _{P\in C}{n_{P}P}$ where $n_{P}\in \mathbb {Z}$ and $n_{p}=0$ for almost all $P$ . The degree of $(D)$ is defined by $\deg(D)=\sum _{P\in C}{c_{P}}$ . $D$ is said to be define over $K$ if $K$ if $D^{\sigma }=\sum _{P\in C}{n_{P}P^{\sigma }=D}$ for all automorphisms σ of ${\overline {K}}$ over $K$ . The set $Div(K)$ of divisor of $C$ defined over K forms an additive abelian group under the addition rule $\sum {a_{p}+b_{p}}=\sum {(a_{p}+b_{p})P}$ . The set $Div^{0}(K)$ of all degree zero divisors of $C$ defined over $K$ is a subgroup of $Div(K)$ .

We take an example:

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

For function $G\in K(C)$ ,the divisor of $G$ is defined by $Div(G)=\sum _{P}{ord_{P}(G)P}$ $G$ can have a pole at point $P$ where $ord_{P}(G)$ is the order of vanishing of $G$ at $P$ . Assume $F,G,H$ are functions over $K$ the divisor of rational function $F=G/H$ is called a principal divisor and is defined by $div(F)=div(G)-div(H)$ . We denote the group of principal divisor $P(K)$ where $P(K)={div(G)|G\in K(C)}$ . The Jacobian of $C$ over $K$ is defined by $J(K)=Div^{0}(K)/P(K)$ . The factor group $J(K)$ is called the divisor class group of $K$ . We denote by ${\overline {D}}\in J(K)$ the class of $D$ in $Div^{0}(K)$ .

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

The desingularization of C has 2 different points at infinity, which we denote $\infty _{1}$ and $\infty _{2}$ . Let $D_{\infty }=\infty _{1}+\infty _{2}$ , note that this divisor is K-rational even if the points $\infty _{1}$ and $\infty _{2}$ are not independently so. if C has a unique point at infinity $\infty$ then $D_{\infty }=d\infty$ , where d is a degree. If d is even and C has two point at infinity $\infty _{1}$ and $\infty _{2}$ then $D_{\infty }={\frac {d}{2}}(\infty _{1}+\infty _{2})$ . If d is odd and C has two points at infinity, then $D_{\infty }={\frac {d+1}{2}}\infty _{1}+{\frac {d-1}{2}}\infty _{2}$ ^[2] For example, let $D_{1}=6P_{1}+4P_{2}$ and $D_{2}=1P_{1}+5P_{2}$ be two divisors, so that the balanced divisor $D_{1}=6P_{1}+4P_{2}-5D_{\infty _{1}}-5D_{\infty _{2}}$ and $D_{2}=1P_{1}+5P_{2}-3D_{\infty _{1}}-3D_{\infty _{2}}$

Transformation From Real Hyperelliptic Curve to Imaginary Hyperelliptic Curve[edit]

Let $C$ be a real quadratic curve over field $K$ . If there exists a ramified prime divisor of degree 1 in $K$ , then we are able to perform a birational transformation to an imaginary quadratic curve.

As we know that divisor $D$ is sum of finite and infinite points $P$ on $C$ , $D=\sum _{P\in C}{n_{P}P}$ where $n_{P}\in \mathbb {Z}$ . A (finite or infinite) point is said to be ramified if it is equal to its own opposite. It means that ${\overline {P}}=(a,-b-h)$ such that $h(a)+2b=0$ . If $P$ is ramified then $n_{p}=1$ where $n_{p}$ is degree of $D$ .^[3]

Any real hyperelliptic curve $C:y^{2}+h(x)y=f(x)$ of genus $g$ with a ramified $K$ - rational finite point $P(a,b)$ is birationally equivalent to an imaginary model $C':y'^{2}+h(x')y'=f(x')$ of genus $g$ , and $K(C)=K(C')$ . ^[4] Here:

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

In our example that $C:y^{2}=f(x)$ where $f(x)=x^{6}+3x^{5}-5x^{4}-15x^{3}+4x^{2}+12x$ h(x) is equal to 0. For any point $P(a,b)$ , $h(a)$ is equal to 0 and $b=0$ . Substituting $h(a)$ and $b$ , we obtain $f(a)=0$ where $f(a)=a(a-1)(a-2)(a+1)(a+2)(a+3)$ .

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

For example, let $a=1$ such that $x={\frac {x'+1}{x'}}$ and $y={\frac {y'}{x'^{3}}}$ , we obtain $\left({\frac {y'}{x'^{3}}}\right)^{2}={\frac {x'+1}{x'}}\left({\frac {x'+1}{x'}}+1\right)\left({\frac {x'+1}{x'}}+2\right)\left({\frac {x'+1}{x'}}+3\right)\left({\frac {x'+1}{x'}}-1\right)\left({\frac {x'+1}{x'}}-2\right)$ . To remove the denominators it is multiplied by $x^{6}$ $y'^{2}=(x'+1)(2x'+1)(3x'+1)(4x'+1)(1)(1-x')$

we get the curve $C':y'^{2}=f(x')$ where $f(x')=(x'+1)(2x'+1)(3x'+1)(4x'+1)(1)(1-x')=-24x'^{5}-26x'^{4}+15x'^{3}+25x'^{2}+9x'+1$ . $C'$ is imaginary quadratic curve since f(x’)has degree $2g+1$ .