Divisors

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

Degree of a divisor as $\mathrm{deg}D={\sum }_{P\in {\mathbb{C}}^{\ast }}{n}_p$.

Divisor of a Meromorphic Function

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

$\text{div}f={\sum }_{P\in {\mathbb{C}}^{\ast }}(or{d}_{P}f)(P)$

It is also known that for any non-zero meromorphic function $f\in K({\mathbb{C}}^{\ast })$, then $\mathrm{deg}\text{div}f=0$.

Example

Let the polynomial be $f(z)=z^2+1$. We have a zero at $i$ and $-i$. But since the domain is ${\mathbb{C}}^{\ast }$, 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, $\text{div}f=(i)+(-i)-2(\infty )$

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:

$$D=\sum _{P\in E({\bar{\mathbb{F}}}_q)}{n}_P(P)$$

This is different than standard group law on curve, which is evident from the notation as the absence of $[\cdot ]$ square brackets around $n_P$ and presence of $(\cdot )$ around $P$.

Example

let $P,Q,R,S\in E({\bar{\mathbb{F}}}_q)$, $D_1=3(P)-2(Q)+1(S)$, and $D_2=(P)+3(Q)+2(S)$, so $\mathrm{deg}(D_1)=3-2+1=2$ and $\mathrm{deg}(D_2)=1+3+2=5$. Calculate $\mathrm{deg}(D_1+D_2)$, $\text{supp}(D_1),\text{supp}(D_2),\text{supp}(D_1+D_2)$.

Another example is taking a function, let’s say a chord $l:\lambda x+\nu$ on $E:y^3=x^3+ax+b$ which gives zeroes on 3 points: $P,Q,-(P\oplus Q)$ with multiplicities $1,1,1$ respectively. Line $l$ also has a pole at curve $E$ at $\mathcal{O}$ with order $3$. Thus, divisor of function $f$, $(l):(P)+(Q)+(-(P+Q))-3(\mathcal{O})$. Degree of divisor: $\mathrm{deg}(l)=0$.

Prerequisites

1. Riemann Sphere: Denoted as ${\mathbb{C}}^{\ast }$ and contains $C\bigcup \{\infty \}$

Resources

• Examples of divisor of a function
• Divisor of a line function on EC
• Divisors and Pairings
• Riemann Sphere