User:Catalina marina

Coordinate ring

The coordinate ring of C over K is defined as

${\displaystyle K[C]={\frac {K[x,y]}{(y^{2}+h(x)y-f(x)}}}$.

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

${\displaystyle {\overline {K}}[C]={\frac {{\overline {K}}[x,y]}{(y^{2}+h(x)y-f(x)}}}$

is an integral domain.
Proof. If r (x,y) were reducible over ${\displaystyle {\overline {K}}}$, it would factor as (y - u(x)) · (y - v(x)) for some u,v${\displaystyle {\overline {K}}}$. 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 ${\displaystyle G(x,y)\in {\overline {K}}}$ can be written uniquely as

${\displaystyle \,G(x,y)=u(x)-v(x)y}$   with ${\displaystyle u(x)}$, ${\displaystyle v(x)}$ ${\displaystyle {\overline {K}}[x]}$

Norm and degree

The conjugate of a polynomial function G(x,y) = u(x) - v(x)y in ${\displaystyle {\overline {K}}[C]}$ is defined to be

${\displaystyle {\overline {G}}(x,y)=u(x)+v(x)(h(x)+y)}$.

The norm of G is the polynomial function ${\displaystyle N(G)=G{\overline {G}}}$. 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

${\displaystyle \,\deg(G)=\max[2\deg(a),2g+1+2\deg(b)]}$.

Properties:

${\displaystyle \;\deg(G)=\deg _{u}(N(G))}$
${\displaystyle \;\deg(GH)=\deg(G)+\deg(H)}$
${\displaystyle \deg(G)=\deg({\overline {G}})}$

The function field K(C) of C over K is the field of fractions of K[C], and the function field ${\displaystyle {\overline {K}}(C)}$ of C over ${\displaystyle {\overline {K}}}$ is the field of fractions of ${\displaystyle {\overline {K}}[C]}$. The elements of ${\displaystyle {\overline {K}}(C)}$ 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

${\displaystyle \,R(P)=G(P)/H(P)}$.

For P a point on C that is not finite, i.e. P = ${\displaystyle \infty }$, we define R(P) as:

If ${\displaystyle \;\deg(G)<\deg(H)}$  then ${\displaystyle R(\infty )=0}$.
If ${\displaystyle \;\deg(G)>\deg(H)}$  then ${\displaystyle R(\infty )}$  is not defined.
If ${\displaystyle \;\deg(G)=\deg(H)}$  then ${\displaystyle R(\infty )}$  is the ratio of the leading coefficients of G and H.

For ${\displaystyle R\in {\overline {K}}(C)^{*}}$ and ${\displaystyle P\in C}$,

If ${\displaystyle \;R(P)=0}$ 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 ${\displaystyle R(P)=\infty }$.

Order of a polynomial function at a point

For ${\displaystyle G{=}u(x)-v(x)\cdot y\in {\overline {K}}[C]^{2}}$ and ${\displaystyle P\in C}$, the order of G at P is defined as:

${\displaystyle \;ord_{P}(G)=r+s}$ 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.
${\displaystyle \;ord_{P}(G)=2r+s}$ if P = (a,b) is a finite Weierstrass point, with r and s as above.
${\displaystyle \;ord_{P}(G)=-deg(G)}$ if P = O.