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2017 Certain K3 surfaces parametrized bythe Fibonacci sequenceviolate the Hasse principle
Dong Quan Ngoc Nguyen
Rocky Mountain J. Math. 47(5): 1693-1710 (2017). DOI: 10.1216/RMJ-2017-47-5-1693

Abstract

For a prime $p \equiv 5 \pmod {8}$ satisfying certain conditions, we show that there exist an infinitude of K3 surfaces parameterized by certain solutions to Pell's equation $X^2 - pY^2 = 4$ in the projective $5$-space that are counterexamples to the Hasse principle explained by the Brauer-Manin obstruction. Further, these surfaces contain no zero-cycle of odd degree over~$\mathbb{Q} $. As an illustration for the main result, we show that the prime $p = 5$ satisfies all of the required conditions in the main theorem, and hence, there exist an infinitude of K3 surfaces parameterized by the Fibonacci sequence that are counterexamples to the Hasse principle explained by the Brauer-Manin obstruction.

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Dong Quan Ngoc Nguyen. "Certain K3 surfaces parametrized bythe Fibonacci sequenceviolate the Hasse principle." Rocky Mountain J. Math. 47 (5) 1693 - 1710, 2017. https://doi.org/10.1216/RMJ-2017-47-5-1693

Information

Published: 2017
First available in Project Euclid: 22 September 2017

MathSciNet: MR3705768
Digital Object Identifier: 10.1216/RMJ-2017-47-5-1693

Subjects:
Primary: 11G35 , 14G05 , 14G25 , 14J28

Keywords: Fibonacci numbers , Hasse principle , K3 surfaces

Rights: Copyright © 2017 Rocky Mountain Mathematics Consortium

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Vol.47 • No. 5 • 2017
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