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Title: Transcendental elements of the Brauer group of diagonal quartic surfaces
Author: Ieronymou, Evis
ISNI:       0000 0001 3586 1466
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2008
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The role played by the Brauer group in the arithmetic of K3 surfaces is not weill understood. Diagonal quartic surfaces is a special class of K3 surfaces, well suited for computations and experimentation as their geometry is well understood. In this thesis, we are exclusively concerned with diagonal quartic surfaces. For a diagonal quartic surface, we have some control over the algebraic part of its Brauer group, but we cannot say much about elements of its Brauer group that do not become trivial when we extend the ground field to an algebraic closure - the so called transcendental elements. We do not even have any explicit descriptions of such elements. The main result of this thesis is that we exhibit the 2-torsion part of the Brauer group of a diagonal quartic surface over an algebraically closed field of characteristic different from 2, in the form of central simple algebras over its function field. This is done in chapter 3: We fix a diagonal quartic surface, S, over the complex numbers. We consider an elliptic fibration of S over the projective line (with no section). We use torsors under groups of multiplicative type over the generic fibre, in order to exhibit elements that span the 2-torsion part of the Brauer group of the generic fibre. Amongst these elements, we find the elements coming from the Brauer group of the surface. Building on our previous results, we give sufficient conditions for the 2-primary part of the Brauer group of a diagonal quartic surface over the rationals, to be algebraic. Moreover, we give an application of a transcendental element. We show that a transcendental element provides an obstruction to weak approximation on a diagonal quartic surface, S, over a number field. We note that. over this number field, the algebraic part of the Brauer group of So is the constant algebras.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available