Elliptic Curves over Q - Elliptic Curves: Number Theory and Cryptography - page 213

Let

x 1 = u 1 1 = 1

u 2 ,
so ( x 1 ,y 1 ) ∈ E ( Q ). We claim that 2( x 1 ,y 1 )= ± ( x, y ).
Equation 8.3 implies that

u 2 ,

u 0 = Au 2 −

u 1 x 1

2 y 1

= A

−

.

2 u 2

Substituting this into (8.1) yields

x = x 1 − 2 Ax 1 − 8 Bx 1 + A 2
4 y 1

.

This is the x -coordinate of 2( x 1 ,y 1 ) (see Theorem 3.6). The y -coordinate is
determined up to sign by the x -coordinate, so 2( x 1 ,y 1 )=( x, ±y )= ± ( x, y ).
It follows that ( x, y )=2( x 1 ,y 1 )or2( x 1 , −y 1 ). In particular, ( x, y ) ∈ 2 E ( Q ).

Example 8.6
We continue with Example 8.5. For the curve y 2 = x ( x − 2)( x +2), we have

φ ( ∞ )=(1 , 1 , 1) , (0 , 0) = ( − 1 , − 2 , 2) ,

φ (2 , 0) = (2 , 2 , 1) , ( − 2 , 0) = ( − 2 , − 1 , 2)

(we used the fact that 4 and 1 are equivalent mod squares to replace 4 by 1).
We eliminated the triple ( a, b, c )=(1 , 2 , 2) by working mod powers of 2. We
now show how to eliminate ( − 1 , − 1 , 1) , (2 , 1 , 2) , ( − 2 , − 2 , 1). Suppose there is
apoint P with φ ( P )=( − 1 , − 1 , 1). Then

φ ( P +(0 , 0)) = φ ( P ) φ (0 , 0) = ( − 1 , − 1 , 1)( − 1 , − 2 , 2) = (1 , 2 , 2) .

But we showed that (1 , 2 , 2) does not come from a point in E ( Q ). Therefore,
P does not exist. The two other triples are eliminated similarly.

Theorem 8.14 has a very important corollary.

THEOREM 8.15 (Weak Mordell-Weil Theorem)
Let E be an elliptic curve defined over Q .Then

E ( Q ) / 2 E ( Q )

is finite.

PROOF We give the proof in the case that e 1 ,e 2 ,e 3 ∈ Q .Asremarked
earlier, we may assume that e 1 ,e 2 ,e 3 ∈ Z .Themap φ in Theorem 8.14 gives