The main result gives sufficient conditions for the existence of a rational point on certain intersections of two quadrics in P<SUP>4</SUP>, assuming two major hypotheses. These assumptions are the finiteness of the Tate-Shafarevitch 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 <I>D</I> of genus 1. We choose these varieties in such a fashion that the Jacobian of thus curve has exactly one rational 2-torsion point. In this situation, the 2-Selmer group of the Jacobian must have at least four elements, one of which is represented by the curve <I>D</I> on the variety. If it has exactly four, then either <I>D</I> has a rational point or it is one of 2 elements of the 2-primary 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 2-Selmer 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 <I>E </I>over Q has a rational point from the conjecture of Birch and Swinnerton-Dyer. It is shown that this assumption implies the existence of an upper bound on the height of a rational point on <I>E. </I>Since it is easily seen that there are only finitely many points with bounded height, this reduces the decision procedure to a finite computation.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:596554 |
| Date | January 2000 |
| Creators | Bender, A. |
| Publisher | University of Cambridge |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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