In this thesis we study at two rather different problems within the arithmetic of abelian varieties. In the first case, we consider the problem of getting uniform bounds for the number of integer points which an elliptic curve defined over Q can obtain within a square box with sides of length N. In particular, our aim is to break the bounds of Bombieri and Pila which in this case give Oe.(N1/3+e). We accomplish this for a large family of elliptic curves using a variety of techniques including repulsion of integer points via Gap Principles, the theory of heights and the Large Sieve. As an application, we prove a result concerning the number of rational points of bounded height on a del Pezzo surface of degree L The second part of the project considers the behaviour of ranks of abelian varieties which arise as Jacobians of curves defined over a number field K. In particular, we ask for which values d are there infinitely many degree d extensions L/K such that the rank of the Jacobian increases as we base change from K to L. In the case of elliptic curves we show that this occurs for every d> 1 and for general Jacobians we show that this holds for all sufficiently large d.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:617932 |
| Date | January 2013 |
| Creators | Mendes da Costa, David John |
| Publisher | University of Bristol |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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