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10 Introduction to Rational Points on Plane Curves

2
The sets (1) and (2) are related by observing that for x = c /4, we have
2
2
[(a + b)/2] = x + A and [(a − b)/2] = x − A.
Hence x, x + A,and x − A are all squares. Conversely, for x as in (2) we define
√
2
2
c = 2 x and a and b with a < b by the requirement that [(a ± b)/2] = c /4 ± A.
Then (a, b, c) is a Pythagorean triple with A = 1/2 ab.
The sets (2) and (3) are related by assuming first that x, x + A,and x − A are
2
2
2
2
4
squares. Then x = u and the product (x + A)(x − A) = x − A = u − A is a
6
2 2
2
2
2
square denoted v . Hence (uv) = u − A u . Setting y = uv and using u = x,we
2
3
2
obtain y = x − A x, i.e., (x, y) is a point on the cubic curve given by the equation
2
2
2
3
y = x − A x. From x = c /4 we see that x is a square with denominator divisible
by 2.
2
2
Conversely, if x = u = (c/2) , i.e., x is a square with denominator divisible
3 2 2 2 2 2 2 2
by 2, and if x − A x is a square y , then v = (y/u) = y /x = x − A =
2
2
2
(x + A)(x − A), and we have a Pythagorean triple v + A = x . The denominators
2
4
2
of x and v are the same t and t is even by assumption. Thus the Pythagorean
2
2
2
2
2
2
triple of integers (t v) + (t A) = (t x) is primitive, and, hence, it is of the form
2
2
2
2
2
2
2
t v = M − N , t A = 2MN,and t x = M + N . By (3.1) this in turn yields a
Pythagorean triple
2 2

2N 2M 2
+ = 4x = (2u)
t t
2
2
2
determining a right triangle of area 2MN/4t = t A/t = A. This establishes the
equivalence between the various sets and proves the proposition.
§4. Rational Cubics and Mordell’s Theorem
Cubics have come up in two places in the previous section. Firstly, there is the Fermat
3
3
cubic x + y = 1 which Euler showed had only two rational points, (1,0) and (0,1).
3
2
2
Secondly, there is the cubic y = x − A x whose rational points tell us about the
existence of right rational triangles of area A. These are special cases of the general
cubic which has the following form in projective coordinates w : x : y:
3 3 3 2 2
0 = c 1 w + c 2 x + c 3 y + c 4 w x + c 5 wx
2 2 2 2
+ c 6 x y + c 7 xy + c 8 w y + c 9 wy + c 10 wxy.
The coefficients are determined only up to a nonzero constant multiple, and, hence,
the cubic is given by c 1 : c 2 : c 3 : c 4 : c 5 : c 6 : c 7 : c 8 : c 9 : c 10 , a point in a nine-
dimensional projective space. This line of ideas is followed further in Chapter 2.
As in the case of conics, our main interest is to describe the rational points on a
rational cuic relative to a given rational point O on the cubic. Again we use a geomet-
ric principle concerning the intersection of a line and a cubic. The difference in this
case is that we do not compare the cubic with another curve as we did for the conic