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.)
Previous Page Next Page