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On the Hasse principle for certain quartic hypersurfaces


Author: Nguyen Ngoc Dong Quan
Journal: Proc. Amer. Math. Soc. 139 (2011), 4293-4305
MSC (2010): Primary 14G05, 11G35, 11G30
DOI: https://doi.org/10.1090/S0002-9939-2011-10936-5
Published electronically: May 4, 2011
MathSciNet review: 2823075
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Abstract: We prove that there are infinitely many non-isomorphic quartic curves which are counter-examples to the Hasse principle explained by the Brauer-Manin obstruction. Further, these quartic curves have no points defined over number fields of odd degree. As a consequence, we show that there are infinitely many quartic hypersurfaces of arbitrary dimension violating the Hasse principle.


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Additional Information

Nguyen Ngoc Dong Quan
Affiliation: Department of Mathematics, The University of Arizona, Tucson, Arizona 85721
Email: dongquan@math.arizona.edu

DOI: https://doi.org/10.1090/S0002-9939-2011-10936-5
Keywords: Azumaya algebras, Brauer groups, Brauer-Manin obstruction, Hasse principle, quartic hypersurfaces
Received by editor(s): November 14, 2009
Received by editor(s) in revised form: July 5, 2010, and October 26, 2010
Published electronically: May 4, 2011
Dedicated: Dedicated to my parents
Communicated by: Ken Ono
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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