Elliptic Curves over Q - Elliptic Curves: Number Theory and Cryptography - page 200

which can be written as

s = t 3 + Ats 2 + Bs 3 .

In the following, it will be convenient to write p j |z for a rational number

z when p j divides the numerator of z . Similarly, we'll write z ≡ 0(mod p j )
in this case. These extended notions of divisibility and congruence satisfy
properties similar to those for the usual notions.

LEMMA 8.3
( x, y ) ∈ E r ifand onlyif p 3 r |s .If p 3 r |s ,then p r |t .

E r ,then p 3 r divides the denominator of y ,so p 3 r PROOF If ( x, y ) ∈

s .Then p 3 r
divides the denominator of y . Part (2) of the theorem shows that p 2 r divides
the denominator of x . Therefore, ( x, y ) ∈ E r .
If p 3 r

Conversely, suppose p 3 r divides the numerator of s =1 /y . |

|s , then the exact power of p dividing the denominator of y is p 3 k ,
with k ≥ r . Part (2) of the theorem implies that the exact power of p dividing
t = x/y is p k .Since k ≥ r ,wehave p r |t .

We now continue with the proof of Theorem 8.1. Let λ r be as in the

statement of the theorem. Note that

p −r x/y =

λ r ( − ( x, y )) = λ r ( x, − y )= − − λ r ( x, y ) .

We now claim that if P 1 + P 2 + P 3 = ∞ then

λ r ( P 1 )+ λ r ( P 2 )+ λ r ( P 3 ) ≡ 0(mod p 4 r ) .

The proof will also show that if P 1 ,P 2 ∈ E r ,then P 3 ∈ E r (hence E r is a
subgroup). Therefore,

λ r ( P 1 + P 2 )= λ r ( − P 3 )= − λ r ( P 3 )= λ r ( P 1 )+ λ r ( P 2 ) ,

so λ r is a homomorphism.
Recall that three points add to ∞ if and only if they are collinear (Exercise
2.6). To prove the claim, let P 1 ,P 2 ,P 3 lie on the line

ax + by + d =0

and assume that P 1 ,P 2 ∈ E r . Dividing by y yields the s, t line

at + b + ds =0 .

Let P i denote the point P i written in terms of the s, t coordinates. In other
words, if

P i =( x i ,y i ) ,