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6 Introduction to Rational Points on Plane Curves

(2.3) Assertion. Assume that L intersects the tangent line T to C at O at a point
R. Then the rational points on the conic C different from the rational point O are
in one-to-one correspondence with the rational points on the line L different from
R. We complete the correspondence between C(Q) and L(Q) by letting O on C
correspond to R on L.
Now we return to the first question of whether there is a rational point at all on a
2
2
2
2
rational conic. For example, clearly the circles x + y = 1and x + y = 2have
2
2
rational points on them. On the other hand, x + y = 3 has no rational point; that
is, it is impossible for the sum of the squares of two rational numbers to equal three.
2
2
To see that there are no rational points on x + y = 3, we can introduce ho-
mogeneous coordinates w : x : y and clear denominators of the rational numbers x
2
2
2
and y to look for integers satisfying x + y = 3w , where x, y, and w have no
2
common factor. In this case 3 does not divide either x or y. For if 3|x,then3|y ,and
2
2
2
hence 3|y. From this it would follow that 9 divides x + y = 3w . This would mean
2
that 3|w and thus 3|w which contradicts the fact that x, y,and w have no common
2
2
factor. This means that x, y ≡±1 (mod 3). This implies that x + y ≡ 1 + 1 = 2
2
2
(mod 3), so that the sum x + y cannot be divisible by 3. We conclude that
2
2
2
x + y = 3w has no solutions. Hence there are no two rational numbers whose
squares add to 3.
2
2
The argument given for x + y = 3 gives an indication of the general method
which can also be applied directly to show that there are no rational points on the
2
2
circle x + y = n for any n of the form n = 4k + 3. The reader is invited to carry
out the argument.
More generally there is a test by which, in a finite numbers of steps, one can
determine whether or not a given rational conic has a rational point. It consists in
seeing whether a certain congruence can be satisfied, and the theorem goes back to
Legendre.
2
2
(2.4) Legendre’s Theorem. For a conic ax + by 2 = w there exists a certain
2
2
2
number m such that ax + by = w has an integral solution if and only if the
congruence
2 2 2
ax + by ≡ w (mod m)
has a solution in the integers modulo m.