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Failure of the Hasse principle on general $K 3$ surfaces

Part of: Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry

Published online by Cambridge University Press:  08 February 2013

Brendan Hassett  and
Anthony Várilly-Alvarado
Brendan Hassett
Affiliation:
Department of Mathematics, Rice University, Houston, TX 77005, USA (hassett@rice.edu; varilly@rice.edu)
Anthony Várilly-Alvarado
Affiliation:
Department of Mathematics, Rice University, Houston, TX 77005, USA (hassett@rice.edu; varilly@rice.edu)
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Abstract

We show that transcendental elements of the Brauer group of an algebraic surface can obstruct the Hasse principle. We construct a general $K 3$ surface $X$ of degree $2$ over $ \mathbb{Q} $, together with a 2-torsion Brauer class $\alpha $ that is unramified at every finite prime, but ramifies at real points of $X$. With motivation from Hodge theory, the pair $(X, \alpha )$ is constructed from a double cover of ${ \mathbb{P} }^{2} \times { \mathbb{P} }^{2} $ ramified over a hypersurface of bidegree $(2, 2)$.

Keywords

Brauer groupsK3 surfacesBrauer–Manin obstructionsquadric bundles

MSC classification

Secondary: 11G35: Varieties over global fields 14G05: Rational points
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 12 , Issue 4 , October 2013 , pp. 853 - 877
Copyright
©Cambridge University Press 2013 

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