Elliptic Curves over Q - Elliptic Curves: Number Theory and Cryptography - page 221

for some constant C . Taking logs yields

h ( P + Q )+ h ( P − Q ) ≤ 2 h ( P )+2 h ( Q )+ c

(8.9)

for some constant c .
We now need to prove the inequality in (8.6). First we'll prove an inequality
between h ( R )and h (2 R )forpoints R .

LEMMA 8.22
Let R ∈ E ( Q ) .Thereexistsaconstant C 2 ,independent of R ,such that

4 h ( R ) ≤ h (2 R )+ C 2 .

PROOF Let

R =( a

b ,y )

with y ∈ Q and a, b ∈ Z with gcd( a, b )=1. Let

h 1 = a 4 − 8 Bab 3 + A 2 b 4

h 2 =(4 b )( a 3 + Aab 2 + Bb 3 )
Δ=4 A 3 +27 B 2 .

− 2 Aa 2 b 2

By Lemma 3.8, there exist homogeneous polynomials r 1 ,r 2 ,s 1 ,s 2 ∈ Z [ a, b ]of
degree 3 (the coecients depend on A, B ) such that

4Δ b 7 = r 1 h 1 + r 2 h 2 (8.10)

4Δ a 7 = s 1 h 1 + s 2 h 2 . (8.11)

For a homogeneous polynomial

p ( x, y )= c 0 x 3 + c 1 x 2 y + c 2 xy 2 + c 3 y 3 ,

we have

) 3 . | p ( a, b ) |≤ ( | c 0 | + | c 1 | + | c 2 | + | c 3 | )Max( | a | , | b |

Suppose | b |≥| a | . It follows that

7

| 4Δ || b | ≤| r 1 ( a, b ) || h 1 | + | r 2 ( a, b ) || h 2 |

3 Max( |h 1 |, |h 2 | ) ,

≤ C 1 |b|

for some constant C 1 independent of R . Therefore,

4

| 4Δ ||b| ≤ C 1 Max( |h 1 |, |h 2 | ) .

Let d =gcd( h 1 ,h 2 ). Then (8.10) and (8.11) imply that

d | 4Δ b 7 and d | 4Δ a 7 .