Hasse's Theorem and Rational Points on the General Conic

Diophantine Equations are equations to which only integer solutions or alternatively rational solutions are considered. The study of these equations has fascinated man for thousands of years. The ancient Babylonians enumerated Pythagorean triples, integer solutions to the equation: X2 + y2 = Z 2 The study of Diophantine equations continued through Greece and the Renaissance with Diophantos, Fermat, Gauss and many others, and up to the present day in which Wiles' very recent proof of Fermat's last theorem still astonishes mathematicians. The 'natural' first case in which the problem of finding solutions to Diophantine equations becomes nontrivial is the conic. Curves are classified by their genus, a certain very useful topological invariant. The Diophantine theory of curves of genus zero is nontrivial but complete. The Diophantine theory of curves of genus greater than zero including elliptic curves is highly nontrivial and incomplete. However, many general theorems exist and the theory is in constant development. In this paper we will detail the theory for conics. We define a point in the plane to be rational if its coordinates are rational. We define a curve to be rational if it is given by an equation with rational coefficients. In this paper we are concerned with finding rational points on rational curves. The first case that one might wish to consider is that of the linear equation. Linear equations pose no problems, especially if rational points are all that is desired, in which case the set of rational points on a rational line (a line with rational coefficients) is obviously in bijective correspondence with the rationals themselves by projection. In the Dilip Das 18.704 Rogalski case of a conic C with a known rational point, say p, the picture is almost as simple. One simply notes that for any other rational point on C, the line passing through it and p is a rational line. Similarly any rational line passing through p must intersect C in a second (not necessarily distinct) rational point in the projective plane, since solving for this intersection point amounts to solving a quadratic equation with one known rational root. In this paper we assume a basic notion of the ideas of projective geometry. For background see the appendix of Silverman and Tate. Thus, we may parameterize all of