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§2. Rational Points on Conics 5

2
0 = f (x, y) = a + bx + cy + dx + exy + fy 2
and in homogeneous form for the projective plane are given by
2 2 2
0 = F(w, x, y) = aw + bwx + cwy + dx + exy + fy .
Observe that the two polynomials are related by f (x, y) = F(1, x, y) and F(w, x, y)
2
= w f (x/w, y/w). More generally, if f (x, y) has degree d, then F(w, x, y) =
d
w f (x/w, y/w) is the corresponding homogenous polynomial, and the curve C f in
x, y-space is the curve C F minus the points on the line at infinity. We will frequently
pass between the projective and affine descriptions of conics and plane curves.
Returning to the conic defined by a polynomial f of degree 2, we begin by ex-
cluding the case where f factors as a product of two linear polynomials, i.e., C f is
the union of two lines or a single double line. These are exactly the singular conics,
and we return later to the general concept of singularity on a curve. One example of
such aconicis xy = 0, the locus for the x and y axis.
(2.1) Remark. Let C = C f be a nonsingular rational conic. There are two questions
related to the determination of the rational points on C:

(1) Is there a rational point P 0 on C at all? If not, then C f (Q) is the empty set!
(2) Given a rational point P 0 on C, determine all other rational points P on C in
terms of P 0 .
The second problem has a particularly simple elegant solution in terms of the
ideas introduced in the previous section. To carry out this solution, we need the fol-
lowing intersection result.
(2.2) Remark. If one of the two intersection points of a rational conic with a rational
line is a rational point, then the other intersection point is rational.
To see this, we use the equation aw + bx + cy = 0 of the line to eliminate one
variable in the second-order equation F(w, x, y) = 0 of the conic. For intersections
off the line at infinity, given by w = 0, one is left with a quadratic equation in the
x coordinate or in the y coordinate of the intersection points. The equation of the
line comes in again here to recover the other coordinate. Thus the intersection points
will be rational if and only if the roots of the quadratic equation are rational. In
general they are conjugate quadratic irrationalities for rational lines and conics, and
an intersection point is rational if and only if its x coordinate is a rational number.
Thus (2.2) reduces to the algebraic statement: if a quadratic polynomial with rational
coefficients has one rational root, then the other root is rational.
Let C be a rational conic with a rational point O on it. Choose a rational line L
not containing O, and project the conic C onto the line L from this point O.
For every point Q on the line L by joining it to O one gets a point P on the
conic C, and in the other direction, a line meets the conic C in two points, so to
every point P on the conic C there corresponds a point Q on the line L. This sets up
a correspondence between points on the conic and points on the line L. Since O is
assumed to be rational, we see from (2.2) that the point P is rational if and only if
the point Q is rational.