Elliptic Curves over Q - Elliptic Curves: Number Theory and Cryptography - page 216

the cube of the denominator of the x -coordinate (when the coe cients A, B
of E are integers), it can be shown that this would change the function h
approximately by a factor of 3/2. This would cause no substantial change in
the theory. In fact, the canonical height h , which will be introduced shortly,
is defined using a limit of values of

1

2 h . It could also be defined as a limit of
values of 1/3 of the height of the y -coordinate. These yield the same canonical
height function. See [109, Lemma 6.3]. The numbers 2 and 3 are the orders
of the poles of the functions x and y on E (see Section 11.1).
It is convenient to replace h with a function h that has slightly better
properties. The function h is called the canonical height .

THEOREM 8.18
Let E be an elliptic curve defined over Q .Thereisafunction

h : E ( Q )

→ R ≥ 0

withthe follow ing properties:
1. h ( P ) ≥ 0 for all P ∈ E ( Q ) .

h ( P ) |≤c 0 for all P .

1

2. T here isaconstant c 0 su ch that |

2 h ( P ) −

3. G iven a constant c ,there are only finitelymanypoints P

∈ E ( Q ) with

h ( P ) ≤ c .

4. h ( mP )= m 2 h ( P ) for allintegers m and all P .

5. h ( P + Q )+ h ( P

Q )=2 h ( P )+2 h ( Q ) for all P, Q . −

6. h ( P )=0 ifand onlyif P isatorsion point.

Property (5) is often called the parallelogram law because if the origin
0 and vectors P, Q, P + Q (ordinary vector addition) are the vertices of a
parallelogram, then the sum of the squares of the lengths of the diagonals
equals the sum of the squares of the lengths of the four sides:

2 + 2 =2 2 +2 2 .

|| P + Q || || P − Q || || P || || Q ||

The proof of Theorem 8.18 will occupy most of the rest of this section. First,
let's use the theorem to deduce the Mordell-Weil theorem.
Proofofthe M ordell-W e iltheorem : Let R 1 ,...,R n be representatives for
E ( Q ) / 2 E ( Q ). Let

h ( R i ) }
and let Q 1 ,...,Q m be the set of points with h ( Q i ) ≤ c . This is a finite set by
Theorem 8.18. Let G be the subgroup of E ( Q ) generated by

c =Max i {

R 1 ,...,R n ,Q 1 ,...,Q m .