44 2. Elliptic curves
Finally, we define the canonical height of P ∈ E(Q) by
h(P ) =
1
2
lim
N→∞
H(2N · P )
4N
.
Note: here
2N
·P means multiplication in the curve, using the addition
law defined in Section 2.4, i.e., 2 · P = P + P ,
22
· P = 2P + 2P , etc.
Example 2.7.2. Let E :
y2
=
x3
+ 877x, and let P be a generator
of E(Q). Here are some values of
1
2
·
H(2N
·P )
4N
:
1
2
· H(P ) = 47.8645312628 . . .
1
2
·
H(2 · P )
4
= 47.7958126219 . . .
1
2
·
H(22
· P )
42
= 47.9720107996 . . .
1
2
·
H(23
· P )
43
= 47.9636902383 . . .
1
2
·
H(24
· P )
44
= 47.9901607777 . . .
1
2
·
H(25
· P )
45
= 47.9901600133 . . .
1
2
·
H(26
· P )
46
= 47.9901569227 . . .
1
2
·
H(27
· P )
47
= 47.9901419861 . . .
1
2
·
H(28 · P )
48
= 47.9901807594 . . . .
The limit is in fact equal to h(P ) = 47.9901859939..., well below the
value
|ΔE|1/2
= 207,773.12....
The canonical height enjoys the following properties and, in fact,
the canonical height is defined so that it is (essentially) the only height
that satisfies these properties:
Proposition 2.7.3 (Néron-Tate). Let E/Q be an elliptic curve and
let h be the canonical height on E.
(1) For all P, Q ∈ E(Q), h(P +Q)+h(P −Q) = 2h(P)+2h(Q).
(Note: this is called the parallelogram law.)
