Definition of divisor of a rational function :

divisor definition

Degree of a divisor as

Divisor of a Meromorphic Function

For a meromorphic function , we can define the divisor of as,

It is also known that for any non-zero meromorphic function , then .

Example

Let the polynomial be . We have a zero at and . But since the domain is , we need to consider point at infinity. In this case, we have a pole at infinity with degree 2 as at .

So, the divisor of f is,

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

  1. Riemann Sphere: Denoted as and contains

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