1 results
Failure of the Hasse principle on general
$K 3$ surfaces
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 12 / Issue 4 / October 2013
- Published online by Cambridge University Press:
- 08 February 2013, pp. 853-877
- Print publication:
- October 2013
-
- Article
- Export citation
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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 
$(2, 2)$.
