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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