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.)