Number Theory: An Introduction to Mathematics

4 Mordell’s Theorem 563

By Proposition II.16,q 3 andq 4 each divideA, and so their product dividesA^2.
Hence, for some integerD=0,

A^2 =Dq 3 q 4 , AC=Dp 3 p 4 , AB=D(p 3 q 4 +p 4 q 3 ).

But it is easily seen thatq 3 q 4 ,p 3 p 4 andp 3 q 4 +p 4 q 3 have no common prime divisor.
It follows thatAdividesD.
Hence, if we put
ρj=max(|pj|,|qj|)( 1 ≤j≤ 4 ),

then

|q 3 q 4 |≤|A|≤ 4 ρ^21 ρ 22 ,

|p 3 p 4 |≤|C|≤[( 1 +|a|)^2 + 8 |b|]ρ^21 ρ 22 ,

|p 3 q 4 +p 4 q 3 |≤|B|≤ 2 ( 1 +|a|+|b|)ρ^21 ρ 22.

But

max(|p 3 |,|q 3 |)max(|p 4 |,|q 4 |)≤max(|p 3 p 4 |,|q 3 q 4 |+|p 3 q 4 +p 4 q 3 |),

since if|q 3 |≤|p 3 |and|p 4 |≤|q 4 |, for example, then

|p 3 q 4 |≤|p 4 q 3 |+|p 3 q 4 +p 4 q 3 |≤|q 3 q 4 |+|p 3 q 4 +p 4 q 3 |.

It follows that there exists a constantC′′>0suchthat

ρ 3 ρ 4 ≤C′′ρ 12 ρ^22 ,

which is equivalent to (∗) withC′=logC′′.

Corollary 11For any P∈E and any integer n,

hˆ(nP)=n^2 hˆ(P).

Proof Sincehˆ(−P)=hˆ(P), we may assumen>0. We may actually assumen>2,
since the result is trivial forn=1 and it holds forn=2 by Proposition 9. By Propo-
sition 10 we have

hˆ(nP)+hˆ((n− 2 )P)= 2 hˆ((n− 1 )P)+ 2 hˆ(P),

from which the general case follows by induction.

It follows from Corollary 11 that if an elementPof the groupEhas finite order,
thenhˆ(P)=0. The converse is also true. In fact, by Proposition 10, the set of all
P∈Esuch thathˆ(P)=0 is a subgroup ofE, and this subgroup is finite since there
are only finitely many pointsPsuch thathˆ(P)<1.
We now deduce from Proposition 10 that a non-negative quadratic form can be
constructed from the canonical height. If we put

(P,Q)=hˆ(P+Q)−hˆ(P)−hˆ(Q),