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Title: Arithmetic of intersections of two quadrics
Author: Bender, A.
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 2000
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The main result gives sufficient conditions for the existence of a rational point on certain intersections of two quadrics in P4, 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 D 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 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 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 E 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 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.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available