Definition of divisor of a rational function $z∈k(C)$:

Degree of a divisor as $gD=∑_{P∈C_{∗}}n_{p}$.

## Divisor of a Meromorphic Function

For a meromorphic function $f(z)$, we can define the **divisor** of $f$ as,

$divf=∑_{P∈C_{∗}}(ord_{P}f)(P)$

It is also known that for any non-zero meromorphic function $f∈K(C_{∗})$, then $gdivf=0$.

Example

Let the polynomial be $f(z)=z_{2}+1$. We have a zero at $i$ and $−i$. But since the domain is $C_{∗}$, we need to consider point at infinity. In this case, we have a pole at infinity with degree 2 as $f(1/z)=0$ at $z=0$. So, the divisor of f is, $divf=(i)+(−i)−2(∞)$

## Divisors of Elliptic Curves

In the case of Riemann sphere, meromorphic functions are considered. In the case of Elliptic curves, rational functions are considered. So, divisor $D$ on $E$ are denoted by multi-set of points on $E$, written as sum:

$D=P∈E(Fˉ_{q})∑ n_{P}(P)$This is different than standard group law on curve, which is evident from the notation as the absence of $[⋅]$ square brackets around $n_{P}$ and presence of $(⋅)$ around $P$.

Example

let $P,Q,R,S∈E(Fˉ_{q})$, $D_{1}=3(P)−2(Q)+1(S)$, and $D_{2}=(P)+3(Q)+2(S)$, so $g(D_{1})=3−2+1=2$ and $g(D_{2})=1+3+2=5$. Calculate $g(D_{1}+D_{2})$, $supp(D_{1}),supp(D_{2}),supp(D_{1}+D_{2})$.

Another example is taking a function, let’s say a chord $l:λx+ν$ on $E:y_{3}=x_{3}+ax+b$ which gives zeroes on 3 points: $P,Q,−(P⊕Q)$ with multiplicities $1,1,1$ respectively. Line $l$ also has a pole at curve $E$ at $O$ with order $3$. Thus, divisor of function $f$, $(l):(P)+(Q)+(−(P+Q))−3(O)$. Degree of divisor: $g(l)=0$.

## Prerequisites

- Riemann Sphere: Denoted as $C_{∗}$ and contains $C⋃{∞}$