28 2. Elliptic curves
Q. Let R be the third point of intersection of L and E. Then R is
also a rational point in E(Q). Let L be the line through R and O.
We define P + Q to be the third point of intersection of L and E.
Notice that any vertical line x = a in the aﬃne plane passes through
[0,1,0], because the same line in projective coordinates is given by
x = az and [0,1,0] belongs to such line. Thus, if E is given by a
+ Ax + B, and O is chosen to be the point [0,1,0],
then L is always a vertical line, so P + Q is always the reflection of
R with respect to the x axis.
The next step in the study of the structure of E(Q) was conjec-
tured by Jules Poincaré in 1908, proved by Louis Mordell in 1922 and
generalized by André Weil in his thesis in 1928:
Theorem 2.4.3 (Mordell-Weil). E(Q) is a finitely generated abelian
group. In other words, there are points P1,...,Pn such that any other
point Q in E(Q) can be expressed as a linear combination
Q = a1P1 + a2P2 + · · · + anPn
for some ai ∈ Z.
The group E(Q) is usually called the Mordell-Weil group of E,
in honor of the two mathematicians who proved the theorem.
Example 2.4.4. Consider the elliptic curve E/Q given by the Weier-
+ y =
− 7x + 6.
The set of rational points E(Q) for this elliptic curve is infinite. For
instance, the following points are on the curve:
(1,0), (2,0), (0, −3), (−3, −1), (8, −22), (−2, −4), (3, −4),
(3,3), (−1, −4), (1, −1), (0,2), (2, −1), (−2,3), (−1,3),
, . . . .
At a first glance, it may seem very diﬃcult to describe all the points
on E(Q), including those listed above, in a succinct manner. However,
the Mordell-Weil theorem tells us that there must be a finite set of
points that generate the whole group. Indeed, it can be proved that