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§5. The Group Law on Cubic Curves and Elliptic Curves 13

P = (... ((P 1 P 2 ) P 3 ) ... P r ) ,

where P 1 ,... , P r are elements of the finite set X with repetitions allowed.
The chord-tangent law of composition is not a group law, because, for example,
there is no identity element, i.e., an element 1 with 1P = P = P1 for all P.However
it does satisfy a commutative law property PQ = QP.

(4.4) Remark. There are infinitely many rational points on a rational line, and there
are either no rational points or infinitely many on a rational conic. The Mordell the-
orem points to a new phenomenon arising with curves in degree 3, namely the possi-
bility of the set of rational points being finite but nonempty. This would be the case
when only a finite number of chord-tangent compositions give all natural points. This
theorem introduces the whole idea of finiteness of number of rational points on a ra-
n
n
tional plane curve. This fits with the Fermat conjecture where x + y = 1has two
points, (1,0) and (0,1), for n odd and four points, (1, 0), (−1, 0), (0, 1),and (0, −1),
for n even where n > 2.
Finally, there is the question of the existence of any rational points on a rational
cubic curve. For conics one could determine by Legendre’s theorem (2.4) in a finite
number of steps, whether a rational conic had a rational point on it or not. For cubics,
there is no known method for determining, in a finite number of steps, whether there
is a rational point. This very important question is still open, and it seems like a very
difficult problem. The idea of looking at the cubic equation over the p-adic numbers
for each prime p is not sufficient in this case, for, in the 1950s, Selmer gave the
example
3
3
3
3x + 4y + 5z = 0.
This is a cubic with a p-adic solution for each p, but with no nontrivial rational
solution. The proof that there is no rational solution is quite a feat.
For the early considerations in this book we will leave aside the problem of the
existence of a rational point and always assume that the cubics we consider have a
given rational point O. Later, in 8 on Galois cohomology, the question of the exis-
tence of a rational point on an auxiliary curve plays a role in estimating the number
of rational points on a given curve with a fixed rational point.

§5. The Group Law on Cubic Curves and Elliptic Curves

It was Jacobi [1835] in Du usu Theoriae Integralium Ellipticorum et Integralium
Abelianorum in Analysi Diophantea who first suggested the use of a group law on
a projective cubic curve. As we have already remarked the chord-tangent law of
composition is not a group law, but with a choice of a rational point O as zero element
and the chord-tangent composition PQ we can define the group law P + Q by the
relation
P + Q = O(PQ).