Number Theory: An Introduction to Mathematics

4 Mordell’s Theorem 567

Moreover, if we define a mapφof the groupEof all rational points ofCA,Binto the
groupE′of all rational points ofCA′,B′by putting

φ(x,y)=(x′,y′) ifx= 0 ,φ(N)=φ(O)=O,

thenφis a homomorphism, i.e.

φ(P+Q)=φ(P)+φ(Q), φ(−P)=−φ(P).

The rangeφ(E)may not be the whole ofE′. In fact, since

x′=(x^3 +Ax^2 +Bx)/x^2 =(y/x)^2 ,

the first coordinate of any finite point ofφ(E)must be a rational square. Furthermore,
ifN=( 0 , 0 )is a point ofφ(E), the integerB′=A^2 − 4 Bmust be a square. We will
show that these conditions completely characterizeφ(E).
Evidently ifA^2 − 4 Bis a square, then the quadratic polynomialX^2 +AX+Bhas
a rational rootx 0 =0andφ(x 0 , 0 )=N. Suppose now that(x′,y′)is a rational point
ofCA′,B′and thatx′=t^2 is a nonzero rational square. We will show that if

x 1 =(t^2 −A+y′/t)/ 2 , y 1 =tx 1 ,

x 2 =(t^2 −A−y′/t)/ 2 , y 2 =−tx 2 ,

then(xj,yj) ∈ Eand φ(xj,yj) = (x′,y′)(j = 1 , 2 ). It is easily seen that
(xj,yj)∈Eif and only if

t^2 =xj+A+B/xj.

But

x 1 x 2 =[(t^2 −A)^2 −y′^2 /t^2 ]/ 4

=[(x′−A)^2 −y′^2 /x′]/ 4

=(x′^3 − 2 Ax′^2 +A^2 x′−y′^2 )/ 4 x′.

Since

y′^2 =x′^3 − 2 Ax′^2 +(A^2 − 4 B)x′,

it follows thatx 1 x 2 =B. Hence(x 1 ,y 1 )and(x 2 ,y 2 )are both inEift^2 =x 1 +A+x 2 ,
and this condition is certainly satisfied by the definitions ofx 1 andx 2.
In addition to

xj+A+B/xj=t^2 =x′(j= 1 , 2 ),

we have

y 1 (x^21 −B)/x^21 =t(x 12 −x 1 x 2 )/x 1 =t(x 1 −x 2 )=y′,