Divisors - Elliptic Curves: Number Theory and Cryptography - page 338

Chapter 11

Divisors

11.1 Definitions and Examples

Let E be an elliptic curve defined over a field K .Foreachpoint P ∈ E ( K ),
define a formal symbol [ P ]. A divisor D on E is a finite linear combination
of such symbols with integer coe cients:
D =
j

a j [ P j ] , j ∈ Z .

A divisor is therefore an element of the free abelian group generated by the
symbols [ P ]. The group of divisors is denoted Div( E ). Define the degree
and sum of a divisor by
deg(
j

a j [ P j ]) =
j

a j ∈ Z

sum(
j

a j [ P j ]) =
j

a j P j ∈ E ( K ) .

The sum function simply uses the group law on E to add up the points that are
inside the symbols. The divisors of degree 0 form an important subgroup of
Div( E ), denoted Div 0 ( E ). The sum function gives a surjective homomorphism

sum : Div 0 ( E ) → E ( K ) .

The surjectivity is because

sum([ P ] − [ ∞ ]) = P.

The kernel consists of divisors of functions (see Theorem 11.2 below), which
we'll now describe.
Assume E is given by y 2 = x 3 + Ax + B .A function on E is a rational
function

f ( x, y ) ∈ K ( x, y )