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

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

# 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 $degdivf=0$.

## 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.

## Prerequisites

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