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Diagonal quartic surfaces and transcendental elements of the Brauer group

Part of: Arithmetic algebraic geometry Diophantine equations

Published online by Cambridge University Press:  28 May 2010

Evis Ieronymou
Evis Ieronymou
Affiliation:
École Polytechnique Fédérale de Lausanne, EPFL-SFB-IMB-CSAG, Station 8, CH-1015, Lausanne, Switzerland (evis.ieronymou@epfl.ch)
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Abstract

We exhibit central simple algebras over the function field of a diagonal quartic surface over the complex numbers that represent the 2-torsion part of its Brauer group. We investigate whether the 2-primary part of the Brauer group of a diagonal quartic surface over a number field is algebraic and give sufficient conditions for this to be the case. In the last section we give an obstruction to weak approximation due to a transcendental class on a specific diagonal quartic surface, an obstruction which cannot be explained by the algebraic Brauer group which in this case is just the constant algebras.

Keywords

diagonal quartic surfacesBrauer groupBrauer–Manin obstructionweak approximation

MSC classification

Secondary: 11G99: None of the above, but in this section 11D25: Cubic and quartic equations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 9 , Issue 4 , October 2010 , pp. 769 - 798
Copyright
Copyright © Cambridge University Press 2010

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