Number Theory: An Introduction to Mathematics

4 Mordell’s Theorem 559

we write simplyE:=E(Q). This section is devoted to the basic theorem of Mordell
(1922), which says thatthe abelian group E is finitely generated.
By replacingXbyX/c^2 andYbyY/c^3 for some nonzeroc∈Q,wemay(and
will) assume thataandbare both integers. LetP=(x,y)be any finite rational point
ofCa,band writex=p/q,wherepandqare coprime integers. Theheight h(P)of
Pis uniquely defined by

h(P)=log max(|p|,|q|).

We also seth(O)=0, whereOis the unique point at infinity ofCa,b.
Evidentlyh(P)≥0. Furthermore,h(−P)=h(P),sinceP =(x,y)implies
−P = (x,−y).Also,foranyr > 0, there exist only finitely many elements
P=(x,y)ofEwithh(P)≤r,sincexdeterminesyup to sign.

Proposition 8There exists a constant C=C(a,b)> 0 such that

|h( 2 P)− 4 h(P)|≤C for all P∈E.

Proof By the formulas given in§3, ifP=(x,y),then2P=(x′,y′),where

x′=m^2 − 2 x, m=( 3 x^2 +a)/ 2 y.

Sincey^2 =x^3 +ax+b, it follows that

x′=(x^4 − 2 ax^2 − 8 bx+a^2 )/ 4 (x^3 +ax+b).

Ifx=p/q,wherepandqare coprime integers, thenx′=p′/q′,where

p′=p^4 − 2 ap^2 q^2 − 8 bpq^3 +a^2 q^4 ,

q′= 4 q(p^3 +apq^2 +bq^3 ).

Evidentlyp′andq′are also integers, but they need not be coprime. However, since

p′=ep′′, q′=eq′′,

wheree,p′′,q′′are integers andp′′,q′′are coprime, we have

h( 2 P)=log max(|p′′|,|q′′|)≤log max(|p′|,|q′|).

Since

max(|p′|,|q′|)≤max(|p|,|q|)^4 max{ 1 + 2 |a|+ 8 |b|+a^2 , 4 ( 1 +|a|+|b|)},

it follows that

h( 2 P)≤ 4 h(P)+C′

for some constantC′=C′(a,b)>0.