\item if $deg(f) > deg(g):~ r ~\text{is not finite at}~ 0$
\item if $deg(f)= deg(g)$ with $deg(f)$ even:\\
$f$'s canonical form leading terms $ax^d$\\
$g$'s canonical form leading terms $bx^d$\\
$a,b \in\Bbbk^\times,~ d=\frac{deg(f)}{2}$, set $r(0)=\frac{a}{b}$
\item if $deg(f)= deg(g)$ with $deg(f)$ odd\\
$f$'s canonical form leading terms $ax^d$\\
$g$'s canonical form leading terms $bx^d$\\
$a,b \in\Bbbk^\times,~ deg(f)=deg(g)=3+2d$, set $r(0)=\frac{a}{b}$
\end{enumerate}
\end{definition}
\subsection{Zeros, poles, uniformizers and multiplicities}
$r \in\Bbbk(E)$ has a \emph{zero} in $P\in E$ if $r(P)=0$\\
$r \in\Bbbk(E)$ has a \emph{pole} in $P\in E$ if $r(P)$ is not finite.
\paragraph{uniformizer:} Let $P\in E$,
uniformizer: rational function $u \in\Bbbk(E)$ with $u(P)=0$ if
$\forall r\in\Bbbk(E)\setminus\{0\},~ \exists d \in\mathbb{Z},~ s\in\Bbbk(E)$ finite at $P$ with $s(P)\neq0$ s.t.
$$r=u^d \cdot s$$
\paragraph{order:} Let $P \in E(\Bbbk)$, let $u \in\Bbbk(E)$ be a uniformizer at $P$.
For $r \in\Bbbk(E)\setminus\{0\}$ being a rational function with $r=u^d \cdot s$ with $s(P)\neq0, \infty$, we say that $r$ has \emph{order}$d$ at $P$ ($ord_P(r)=d$).
\paragraph{multiplicity:}\emph{multiplicity of a zero} of $r$ is the order of $r$ at that point, \emph{multiplicity of a pole} of $r$ is the order of $r$ at that point.
if $P \in E(\Bbbk)$ is neither a zero or pole of $r$, then $ord_P(r)=0$ ($=d,~ r=u^0s$).
\vspace{0.5cm}
\begin{minipage}{4.3 in}
\paragraph{Multiplicities, from the book "Elliptic Tales"} (p.69), to provide intuition
Factorization into \emph{linear factors}: $p(x)=c\cdot(x-a_1)\cdots(x-a_d)$\\
$d$: degree of $p(x)$, $a_i \in\Bbbk$\\
Solutions to $p(x)=0$ are $x=a_1, \ldots, a_d$ (some $a_i$ can be repeated)\\
eg.: $p(x)=(x-1)(x-1)(x-3)$, solutions to $p(x)=0:~ 1, 1, 3$\\
$x=1$ is a solution to $p(x)=0$ of \emph{multiplicity} 2.
The total number of solutions (counted with multiplicity) is $d$, the degree of the polynomial whose roots we are finding.
\end{minipage}
\section{Divisors}
\begin{definition}{Divisor}
$$D=\sum_{P \in E(\mathbb{K})} n_p \cdot[P]$$
$$D=\sum_{P \in E(\Bbbk)} n_p \cdot[P]$$
\end{definition}
\begin{definition}{Degree \& Sum}
$$deg(D)=\sum_{P \in E(\mathbb{K})} n_p$$
$$sum(D)=\sum_{P \in E(\mathbb{K})} n_p \cdot P$$
$$deg(D)=\sum_{P \in E(\Bbbk)} n_p$$
$$sum(D)=\sum_{P \in E(\Bbbk)} n_p \cdot P$$
\end{definition}
The set of all divisors on $E$ forms a group: for $D =\sum_{P\in E(\Bbbk)} n_P[P]$ and $D' =\sum_{P\in E(\Bbbk)} m_P[P]$,
$$D+D' =\sum_{P\in E(\Bbbk)}(n_P + m_P)[P]$$
\begin{definition}{Associated divisor}
$$div(r)=\sum_{P \in E(\Bbbk)} ord_P(r)[P]$$
\end{definition}
Observe that
\begin{enumerate}
\item[]$div(rs)= div(r)+div(s)$
\item[]$div(\frac{r}{s})= div(r)-div(s)$
\end{enumerate}
Observe that
$$\sum{P \in E(\Bbbk)} ord_P(r)\cdot P =0$$
\begin{definition}{Support}
$$\sum_P n_P[P], ~\forall P \in E(\Bbbk)\mid n_P \neq0$$
\end{definition}
\begin{definition}{Principal divisor}
@ -62,21 +163,21 @@
$D \sim D'$ iff $D - D'$ is principal.
\begin{definition}{Evaluation of a rational function}
\begin{definition}{Evaluation of a rational function} (function $r$ evaluated at $D$)
$$r(D)=\prod r(P)^{n_p}$$
\end{definition}
\section{Weil reciprocity}
\begin{theorem}{(Weil reciprocity)}
Let $E/\mathbb{K}$ be an e.c. over an alg. closed field. If $r,~s \in\mathbb{K}\setminus\{0\}$ are rational functions whose divisors have disjoint support, then
Let $E/\Bbbk$ be an e.c. over an alg. closed field. If $r,~s \in\Bbbk\setminus\{0\}$ are rational functions whose divisors have disjoint support, then
$$r(div(s))= s(div(r))$$
\end{theorem}
Proof. (todo)
\section{Generic Weil Pairing}
Let $E(\mathbb{K})$, with $\mathbb{K}$ of char $p$, $n$ s.t. $p \nmid n$.
Let $E(\Bbbk)$, with $\Bbbk$ of char $p$, $n$ s.t. $p \nmid n$.
$\mathbb{K}$ large enough: $E(\mathbb{K})[n]= E(\mathbb{\overline{K}})=\mathbb{Z}_n \oplus\mathbb{Z}_n$ (with $n^2$ elements).
$\Bbbk$ large enough: $E(\Bbbk)[n]= E(\overline{\Bbbk})=\mathbb{Z}_n \oplus\mathbb{Z}_n$ (with $n^2$ elements).
