Divisors - Elliptic Curves: Number Theory and Cryptography - page 341

Let's compute the order of x and y . Rewriting the equation for E as
x
y

2

= x − 1 1+ A

x 3 − 1

B

x 2 +

shows that x/y vanishes at ∞ and that

ord ∞ ( x )= − 2

(given that x/y is a uniformizer). Since y = x · ( y/x ), we have

ord ∞ ( y )= − 3 .

Note that the orders of x and y at ∞ agree with what we expect from looking

at the Weierstrass ℘ -function.

If f is a function on E that is not identically 0, define the divisor of f to

be

div( f )=
P

ord P ( f )[ P ] ∈ Div( E ) .

∈ E ( K )

This is a finite sum, hence a divisor, by the following.

PROPOSITION 11.1
Let E be an ellipticcurveand et f be a function on E that isnotidentically
0.

1. f has only finitely m any zeros and poles

2. deg(div( f )) = 0

3. If f has no zeros or poles (so div( f )=0 ), then f isaconstant.

For a proof, see [42, Ch.8, Prop. 1] or [49, II, Cor. 6.10]. The complex
analytic analogue of the proposition is T heorem 9.1. Note that it is important
to look at points with coordinates in K . It is easy to construct nonconstant
functions with no zeros or poles at the points in E ( K ), and it is easy to
construct functions that have zeros but no poles in E ( K ) (see Exercise 11.1).
The divisor of a function is said to be a principal divisor .
Suppose P 1 ,P 2 ,P 3 are three points on E that lie on the line ax + by + c =0.
Then the function

f ( x, y )= ax + by + c

has zeros at P 1 ,P 2 ,P 3 .If b =0then f has a triple pole at ∞ . Therefore,

div( ax + by + c )=[ P 1 ]+[ P 2 ]+[ P 3 ] − 3[ ∞ ] .