Title:

Arithmetic of intersections of two quadrics

The main result gives sufficient conditions for the existence of a rational point on certain intersections of two quadrics in P^{4}, assuming two major hypotheses. These assumptions are the finiteness of the TateShafarevitch group III of an elliptic curve over Q and the Schinzel Hypothesis. The intersection of the respective variety with a hyperplane in general position is a curve D of genus 1. We choose these varieties in such a fashion that the Jacobian of thus curve has exactly one rational 2torsion point. In this situation, the 2Selmer group of the Jacobian must have at least four elements, one of which is represented by the curve D on the variety. If it has exactly four, then either D has a rational point or it is one of 2 elements of the 2primary component of III. By a theorem of Cassels, this is impossible if we assume III to be finite. The Schinzel Hypothesis is needed to carry out the explicit calculations to derive conditions for the 2Selmer group to have exactly four elements. The fourth chapter supplies the details to a sketch of Y.I. Manin. It derives an algorithm to decide whether an arbitrary elliptic curve E over Q has a rational point from the conjecture of Birch and SwinnertonDyer. It is shown that this assumption implies the existence of an upper bound on the height of a rational point on E. Since it is easily seen that there are only finitely many points with bounded height, this reduces the decision procedure to a finite computation.
