Elliptic Curves over Q - Elliptic Curves: Number Theory and Cryptography - page 215

THEOREM 8.17 (Mordell-Weil)
Let E be an elliptic curve defined over Q .Then E ( Q ) is a finitely generated
abelian group.

The theorem says that there is a finite set of points on E from which all
other points can be obtained by repeatedly drawing tangent lines and lines
through points, as in the definition of the group law. The proof will be given
below. Since we proved the weak Mordell-Weil theorem only in the case that
E [2] ⊆ E ( Q ), we obtain the theorem only for this case. However, the weak
Mordell-Weil theorem is true in general, and the proof of the passage from
the weak result to the strong result holds in general.
From the weak Mordell-Weil theorem, we know that E ( Q ) / 2 E ( Q )isfi-
nite. This alone is not enough to deduce the stronger result. For example,
R / 2 R = 0, hence is finite, even though R is not finitely generated. In our
case, suppose we have points R 1 ,...,R n representing the finitely many cosets
in E ( Q ) / 2 E ( Q ). Let P ∈ E ( Q ) be an arbitrary point. We can write

P = R i +2 P 1

for some i and some point P 1 .Thenwewrite

P 1 = R j +2 P 2 ,

etc. If we can prove the process stops, then we can put things back together
and obtain the theorem. The theory of heights will show that the points
P 1 ,P 2 ,... are getting smaller, in some sense, so the process will eventually
yield a point P k that lies in some finite set of small points. These points, along
with the R i , yield the generators of E ( Q ). We make these ideas more precise
after Theorem 8.18 below. Note that sometimes the points R i by themselves
do not suce to generate E ( Q ). See Exercise 8.7.
Let a/b be a rational number, where a, b are integers with gcd( a, b )=1.
Define

H ( a/b )=Max( |a|, |b| )

and

h ( a/b )=log H ( a/b ) .

The function h is called the (logarithmic) height function . It is closely
related to the number of digits required to write the rational number a/b .
Note that, given a constant c , there are only finitely many rational numbers
x with h ( x ) c .

Now let E be an elliptic curve over Q and let ( x, y ) ∈ E ( Q ). Define

≤

h ( x, y )= h ( x ) , ( ∞ )=0 , ( x, y )= H ( x ) , H ( ∞ )=1 .

It might seem strange using only the x -coordinate. Instead, we could use
the y -coordinate. Since the square of the denominator of the y -coordinate is