If the descent theory, developed by Colliot-Thelene and Sansuc, applies, then it can reduce the question of understanding whether the Brauer-Manin obstruction is the only one to understanding weak approximation on the resulting descent varieties. In some cases the descent varieties are easier to handle and accessible by analytic methods as for example t he circle method. In joint work with A. Skorobogatov we followed this approach focusing on varieties corresponding to the representation of a norm form by a product of linear polynomials. We present this work in the first part of this thesis which involves an application of the circle method over number fields to systems of linear equations involving norm forms. In the second part of this thesis we study the arithmetic of subvarieties in biprojective space. So far, the circle method has been a very useful tool to prove many cases of Manin's conjecture. Work of B. Birch back in 1962 establishes this for smooth complete intersections in projective space as soon as the number of variables is large enough depending on the degree and number of equations. In biprojective space there is not much known so far, unless the underlying polynomials are of bidegree (1,1). A combination of the circle method with the generalised hyperbola method recently developed by V. Blomer and J. Brudern allows us to verify Manin 's conjecture for certain smooth hypersurfaces of general bidegree in biprojective space.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:601162 |
| Date | January 2013 |
| Creators | Schindler, Damaris |
| Publisher | University of Bristol |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
Page generated in 0.0019 seconds