Number Theory: An Introduction to Mathematics

568 XIII Connections with Number Theory

and similarlyy 2 (x 22 −B)/x 22 =y′. It follows that

φ(x 1 ,y 1 )=φ(x 2 ,y 2 )=(x′,y′).

Sinceφis a homomorphism, the rangeφ(E)is a subgroup ofE′. We are going to
show that this subgroup is of finite index inE′. By what we have already proved for
E, there exists a finite (or empty) setP 1 ′=(x 1 ′,y 1 ′),...,Ps′=(x′s,y′s)of points ofE′
such thatxi′is not a rational square( 1 ≤i≤s)andsuchthat,ifP′=(x′,y′)is any
other point ofE′for whichx′is not a rational square, thenx′x′jis a nonzero rational
square for a uniquej∈{ 1 ,...,s}.LetP′′=(x′′,y′′)be the third point of intersection
withCA′,B′of the line throughP′andP′j,sothat

P′+P′j+P′′=O.

By what we have already proved, eitherx′′is a nonzero rational square orP′′=Nand
x′x′j=B′is a square. In either case,P′′∈φ(E).Furthermore,if2P′j=(x ̄,− ̄y),then

eitherx ̄is a nonzero rational square or 2P′j=Nandx′j^2 =B′. In either case again,

2 P′j∈φ(E).Since

P′=P′j−( 2 Pj′+P′′),

it follows thatP′andP′jare in the same coset ofφ(E). ConsequentlyP 1 ′,...,Ps′,

together withO,andalsoNifB′is not a square, form a complete set of representa-
tives of the cosets ofφ(E)inE′.
The preceding discussion can be repeated withCA′,B′in the place ofCA,B. It yields
a homomorphismφ′of the groupE′of all rational points ofCA′,B′into the groupE′′
of all rational points ofCA′′,B′′,where

A′′=− 2 A′= 4 A, B′′=A′^2 − 4 B′= 16 B.

But the simple transformation(X,Y)→(X/ 4 ,Y/ 8 )replacesCA′′,B′′byCA,Band
defines an isomorphismχofE′′withE. Hence the composite mapψ=χ◦φ′is a
homomorphism ofE′intoE,andψ◦φis a homomorphism ofEinto itself.
We now show that the homomorphismP→ψ◦φ(P)is just the doubling map
P→ 2 P. Since this is obvious ifP=OorN, we need only verify it forP=(x,y)
withx=0.
ForP′′=φ′◦φ(P)we have

x′′=(y′/x′)^2 =[y( 1 −B/x^2 )·x^2 /y^2 ]^2 =(x^2 −B)^2 /y^2

and

y′′=y′( 1 −B′/x′^2 )=y( 1 −B/x^2 )[1−(A^2 − 4 B)x^4 /y^4 ]

=(x^2 −B)[y^4 −(A^2 − 4 B)x^4 ]/x^2 y^3

=(x^2 −B)[(x^2 +Ax+B)^2 −(A^2 − 4 B)x^2 ]/y^3.