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).