User:Catalina marina

Coordinate ring
The coordinate ring of $C$ over $K$ is defined as
 * $$K[C]=\frac{K[x,y]}{(y^2+h(x)y-f(x)}$$.

The polynomial $$\,r(x,y)=y^2+h(x)y-f(x)$$ is irreducible over $$\overline{K}$$, so
 * $$\overline{K}[C]=\frac{\overline{K}[x,y]}{(y^2+h(x)y-f(x)}$$

is an integral domain. Proof. If $r (x,y)$ were reducible over $$\overline{K}$$, it would factor as $(y - u(x))(y - v(x))$ for some $u,v$ &isin; $$\overline{K}$$. But then $u(x)v(x)= f(x)$ so it has degree $2g + 1$, and $a(x) + b(x) = h(x)$ so it has degree smaller than $g$, which is impossible.

Note that any polynomial function $$G(x,y)\in\overline{K}$$ can be written uniquely as
 * $$\,G(x,y)=u(x)-v(x)y$$ with $$u(x)$$, $$v(x)$$'' &isin; $$\overline{K}[x]$$

Norm and degree
The conjugate of a polynomial function $G(x,y) = u(x) - v(x)y$ in $$\overline{K}[C]$$ is defined to be
 * $$\overline{G}(x,y)=u(x)+v(x)(h(x)+y)$$.

The norm of $G$ is the polynomial function $$N(G)=G\overline{G}$$. Note that $N(G) = u(x)^{2} + u(x)v(x)h(x) - v(x)^{2}f(x)$, so $N(G)$ is a polynomial in only one variable.

If $G(x,y) = u(x) - v(x)y$, then the degree of $G$ is defined as
 * $$\,\deg(G)=\max[2\deg(a),2g+1+2\deg(b)]$$.

Properties:
 * $$\;\deg(G)=\deg_u(N(G))$$
 * $$\;\deg(GH)=\deg(G)+\deg(H)$$
 * $$\deg(G)=\deg(\overline{G})$$

The function field $K(C)$ of $C$ over $K$ is the field of fractions of $K[C]$, and the function field $$\overline{K}(C)$$ of $C$ over $$\overline{K}$$ is the field of fractions of $$\overline{K}[C]$$. The elements of $$\overline{K}(C)$$ are called rational functions on $C$. For $R$ such a rational function, and $P$ a finite point on $C$, $R$ is said to be defined at $P$ if there exist polynomial functions $G, H$ such that $R = G/H$ and $H(P) &ne; 0$, and then the value of $R$ at $P$ is
 * $$\,R(P)=G(P)/H(P)$$.

For $P$ a point on $C$ that is not finite, i.e. $P$ = $$\infin$$, we define $R(P)$ as:
 * If $$\;\deg(G)<\deg(H)$$ then $$R(\infin)=0$$.
 * If $$\;\deg(G)>\deg(H)$$ then $$R(\infin)$$ is not defined.
 * If $$\;\deg(G)=\deg(H)$$ then $$R(\infin)$$ is the ratio of the leading coefficients of $G$ and $H$.

For $$R\in\overline{K}(C)^*$$ and $$P\in C$$,
 * If $$\;R(P)=0$$ then $R$ is said to have a zero at $P$,
 * If $R$ is not defined at $P$ then $R$ is said to have a pole at $P$, and we write $$R(P)=\infin$$.

Order of a polynomial function at a point
For $$G = u(x) - v(x)\cdot y\in\overline{K}[C]^2$$ and $$P\in C$$, the order of $G$ at $P$ is defined as:
 * $$\;ord_P(G)=r+s$$ if $P = (a,b)$ is a finite point which is not Weierstrass. Here $r$ is the highest power of $(x-a)$ which divides both $u(x)$ and $v(x)$. Write $G(x,y) = (x - a)^{r}(u_{0}(x) - v_{0}(x)y)$ and if $u_{0}(a) - v_{0}(a)b = 0$, then $s$ is the highest power of $(x - a)$ which divides $N(u_{0}(x) - v_{0}(x)y = u_{0}^{2} + u_{0}v_{0}h - v_{0}^{2}f$, otherwise, $s = 0$.
 * $$\;ord_P(G)=2r+s$$ if $P = (a,b)$ is a finite Weierstrass point, with $r$ and $s$ as above.
 * $$\;ord_P(G)=-deg(G)$$ if $P = O$.