566 XIII Connections with Number Theory
This shows that each prime which dividesm, but notm^2 +Ame^2 +Be^4 , must occur
to an even power inm. On the other hand, each prime which divides bothmand
m^2 +Ame^2 +Be^4 must also divideB,since(m,e)=1. Consequently we can write
x=±pε 11 ···p
εk
k(u/e)(^2) ,
whereu ∈ N, p 1 ,...,pk are the distinct primes dividingB andεj ∈{ 0 , 1 }
( 1 ≤j ≤k). Hence there are at most 2k+^1 rational square classes to whichxcan
belong.
Suppose now thatP 1 =(x 1 ,y 1 )andP 2 =(x 2 ,y 2 )are distinct rational points of
CA,Bfor whichx 1 x 2 is a nonzero rational square, and letP 3 =(x 3 ,y 3 )be the third
point of intersection withCA,Bof the line throughP 1 andP 2 .Thenx 1 ,x 2 ,x 3 are the
three roots of a cubic equation
(mX+c)^2 =X^3 +AX^2 +BX.
From the constant term we see thatx 1 x 2 x 3 =c^2. It follows thatx 3 is a nonzero
rational square ifc=0. Ifc=0, thenP 3 =Nandx 1 x 2 =B.
Suppose next thatP=(x,y)is any rational point ofCA,Bwithx=0, and let
2 P=(x ̄,− ̄y).ThenP ̄=(x ̄,y ̄)is the other point of intersection withCA,Bof the
tangent toCA,BatP. By the same argument as before,x^2 x ̄=c^2. Hencex ̄is a nonzero
rational square ifc=0. Ifc=0, then 2P=Nandx^2 =B.
To deduce thatE/ 2 Eis finite from these observations we will use an arithmetic
analogue of Landen’s transformation. We saw in Chapter XII that, over the fieldCof
complex numbers, the cubic curveCλdefined by the polynomialY^2 −gλ(X),where
gλ(X)= 4 X( 1 −X)( 1 −λX), admits the parametrization
X=S(u,λ), Y=S′(u,λ).
It follows from Proposition XII.11 that the cubic curveCλ′,whereλ′is given by
λ′=λ^2 /[1+( 1 −λ)^1 /^2 ]^4 , admits the parametrization
X′=[1+( 1 −λ)^1 /^2 ]X( 1 −X)/( 1 −λX),
Y′=[1+( 1 −λ)^1 /^2 ]Y( 1 − 2 X+λX^2 )/( 1 −λX)^2 ,
where againX=S(u,λ),Y=S′(u,λ)and where( 1 − 2 X+λX^2 )/( 1 −λX)^2 is
the derivative with respect toXofX( 1 −X)/( 1 −λX). Since alsoX′=S(u′,λ′),
whereu′=[1+( 1 −λ)^1 /^2 ]u,themap(X,Y)→(X′,Y′)defines a homomorphism
of the group of complex points ofCλinto the group of complex points ofCλ′.
We will simply state analogous results for the cubic curveCA,Bover the fieldQof
rational numbers, since their verification is elementary. If(x,y)is a rational point of
CA,Bwithx=0andif
x′=(x^2 +Ax+B)/x, y′=y(x^2 −B)/x^2 ,
then(x′,y′)is a rational point ofCA′,B′,where
A′=− 2 A, B′=A^2 − 4 B.