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 ∈ ${\overline {K}}$. 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 $u(x)$, $v(x)$ ∈ ${\overline {K}}[x]$

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)²f(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) ≠ 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 = $\infty$, we define R(P) as:

If $\;\deg(G)<\deg(H)$  then $R(\infty )=0$.

If $\;\deg(G)>\deg(H)$  then $R(\infty )$  is not defined.

If $\;\deg(G)=\deg(H)$  then $R(\infty )$  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 $R(P)=\infty$.

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(u₀(x) - v₀(x)y) and if u₀(a) - v₀(a)b = 0, then s is the highest power of (x - a) which divides N(u₀(x) - v₀(x)y = u₀² + u₀v₀h - v₀²f, 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.