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Mathematics of Computation

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Odd order obstructions to the Hasse principle on general K3 surfaces

Authors: Jennifer Berg and Anthony Várilly-Alvarado
Journal: Math. Comp. 89 (2020), 1395-1416
MSC (2010): Primary 14J28, 14G05, 14J35, 14F22
Published electronically: November 12, 2019
MathSciNet review: 4063322
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Abstract: We show that odd order transcendental elements of the Brauer group of a K3 surface can obstruct the Hasse principle. We exhibit a general K3 surface $ Y$ of degree 2 over $ \mathbb{Q}$ together with a $ 3$-torsion Brauer class $ \alpha $ that is unramified at all primes except for $ 3$, but ramifies at all $ 3$-adic points of $ Y$. Motivated by Hodge theory, the pair $ (Y, \alpha )$ is constructed from a cubic fourfold $ X$ of discriminant 18 birational to a fibration into sextic del Pezzo surfaces over the projective plane. Notably, our construction does not rely on the presence of a central simple algebra representative for $ \alpha $. Instead, we prove that a sufficient condition for such a Brauer class to obstruct the Hasse principle is insolubility of the fourfold $ X$ (and hence the fibers) over $ \mathbb{Q}_3$ and local solubility at all other primes.

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Additional Information

Jennifer Berg
Affiliation: Department of Mathematics, 380 Olin Science Building, Bucknell University, Lewisburg, Pennsylvania 17837

Anthony Várilly-Alvarado
Affiliation: Department of Mathematics, Rice University MS 136, Houston, Texas 77005-1892

Keywords: K$3$ surface, cubic fourfolds, Hasse principle, Brauer--Manin obstruction
Received by editor(s): March 3, 2019
Received by editor(s) in revised form: July 23, 2019
Published electronically: November 12, 2019
Additional Notes: The second author was partially supported by NSF grants DMS-1352291 and DMS-1902274
Article copyright: © Copyright 2019 American Mathematical Society