2.7. The rank and the free part of E(Q) 43 2.7. The rank and the free part of E(Q) In the previous sections we have described simple algorithms that de- termine the torsion subgroup of E(Q). Recall that the Mordell-Weil theorem (Thm. 2.4.3) says that there is a (non-canonical) isomor- phism E(Q) E(Q)torsion ZRE. Our next goal is to try to find RE generators of the free part of the Mordell-Weil group. Unfortunately, no algorithm is known that will always yield such free points. We don’t even have a way to determine RE, the rank of the curve, although sometimes we can obtain upper bounds for the rank of a given curve E/Q (see, for instance, Theorem 2.7.4 below). Naively, one could hope that if the coefficients of the (minimal) Weierstrass equation for E/Q are small, then the coordinates of the generators of E(Q) should also be small, and perhaps a brute force computer search would yield these points. However, Bremner and Cassels found the following surprising example: the curve y2 = x3 + 877x has rank equal to 1 and the x-coordinate of a generator P is x(P ) = (612776083187947368101/78841535860683900210)2. However, Serge Lang salvaged this idea and conjectured that for all 0 there is a constant C such that there is a system of generators {Pi : i = 1,...,RE} of E(Q) with h(Pi) C · |ΔE|1/2+ , where h is the canonical height function of E/Q, which we define next. Lang’s conjecture says that the size of the coordinates of a generator may grow exponentially with the (minimal) discriminant of a curve E/Q. Definition 2.7.1. We define the height of m n Q, with gcd(m, n) = 1, by h m n = log(max{|m|, |n|}). This can be used to define a height on a point P = (x, y) on an elliptic curve E/Q, with x, y Q by H(P ) = h(x).
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