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On the Frequency of Algebraic Brauer Classes on Certain Log K3 Surfaces

Part of: Arithmetic problems. Diophantine geometry Surfaces and higher-dimensional varieties Forms and linear algebraic groups

Published online by Cambridge University Press:  03 December 2018

Jörg Jahnel  and
Damaris Schindler
Jörg Jahnel
Affiliation:
Department Mathematik, Univ. Siegen, Walter-Flex-Str. 3, D-57068 Siegen, Germany Email: jahnel@mathematik.uni-siegen.de URL: http://www.uni-math.gwdg.de/jahnel
Damaris Schindler
Affiliation:
Mathematisch Instituut, Universiteit Utrecht, Budapestlaan 6, NL-3584 CD Utrecht, The Netherlands Email: d.schindler@uu.nl URL: http://www.uu.nl/staff/DSchindler
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Abstract

Given systems of two (inhomogeneous) quadratic equations in four variables, it is known that the Hasse principle for integral points may fail. Sometimes this failure can be explained by some integral Brauer–Manin obstruction. We study the existence of a non-trivial algebraic part of the Brauer group for a family of such systems and show that the failure of the integral Hasse principle due to an algebraic Brauer–Manin obstruction is rare, as for a generic choice of a system the algebraic part of the Brauer-group is trivial. We use resolvent constructions to give quantitative upper bounds on the number of exceptions.

Keywords

Brauer classesBrauer–Manin obstructionlog K3 surfaces

MSC classification

Primary: 11E12: Quadratic forms over global rings and fields
Secondary: 14G20: Local ground fields 14G25: Global ground fields 14J20: Arithmetic ground fields
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 3 , September 2019 , pp. 551 - 563
Copyright
© Canadian Mathematical Society 2018 

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Footnotes

The second author was supported by the NWO Veni Grant No. 016.Veni.173.016.

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