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.)

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2011 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.