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COUNTEREXAMPLES TO THE HASSE PRINCIPLE IN FAMILIES

Part of: Diophantine equations

Published online by Cambridge University Press:  05 November 2021

NGUYEN XUAN THO Open the ORCID record for NGUYEN XUAN THO [Opens in a new window]
NGUYEN XUAN THO*
Affiliation:
School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Hanoi, Vietnam
*
e-mail: [email protected]
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Abstract

We generalise two quartic surfaces studied by Swinnerton-Dyer to give two infinite families of diagonal quartic surfaces which violate the Hasse principle. Standard calculations of Brauer–Manin obstructions are exhibited.

Keywords

Brauer–Manin obstructionHasse principlediagonal quartic surfaces

MSC classification

Primary: 11D25: Cubic and quartic equations
Secondary: 11D88: $p$-adic and power series fields
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 106 , Issue 1 , August 2022 , pp. 18 - 27
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The author is supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) (grant number 101.04-2019.314).

References

Bremner, A. and Tho, N. X., ‘An interesting quartic surface, everywhere locally solvable, with cubic point but no global point’, Publ. Math. Debrecen 93(1–2) (2018), 253–260.CrossRefGoogle Scholar
Bright, M., Computations on Diagonal Quartic Surfaces (PhD Dissertation, University of Cambridge, 2002).Google Scholar
Bright, M., ‘Brauer groups of diagonal quartic surfaces’, J. Symbolic Comput. 41 (2006), 544–558.Google Scholar
Bright, M., ‘The Brauer–Manin obstruction on a general diagonal quartic surface’, Acta Arith. 147(3) (2021), 291–302.CrossRefGoogle Scholar
Cassels, J. W. S. and Guy, M. J. T., ‘On the Hasse principle for cubic surfaces’, Mathematika 13 (1966), 111–120.CrossRefGoogle Scholar
Colliot-Thélène, J.-L., Kanevsky, D. and Sansuc, J.-J., ‘Arithmétique des surfaces cubiques diagonales’, in: Diophantine Approximation and Transcendence Theory (Bonn, 1985) (ed. G. Wüstholz), Lecture Notes in Mathematics, 1290 (Springer, Berlin, 1987), 1–108.Google Scholar
Cox, D. A., Primes of the Form ${x}^2+n{y}^2$ : Fermat, Class Field Theory and Complex Multiplication, 2nd edn (John Wiley, Hoboken, NJ, 2013).Google Scholar
Hirakawa, Y., ‘Counterexamples to the local-global principle associated with Swinnerton-Dyer’s cubic form’, Rocky Mountain J. Math. 50(6) (2020), 2097–2102.CrossRefGoogle Scholar
Lemmermeyer, F., Reciprocity Laws: From Euler to Eisenstein (Springer, Berlin, 2000).Google Scholar
Lind, C. E., Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins (Thesis, University of Uppsala, 1940).Google Scholar
Manin, Y. I., Cubic Forms: Algebra, Geometry, Arithmetic (North Holland, Amsterdam, 2012).Google Scholar
Poonen, B., ‘An explicit family of genus-one curves violating the Hasse principle’, J. Théor. Nombres Bordeaux 13(1) (2001), 263–274.CrossRefGoogle Scholar
Quan, N. N. D., ‘On the Hasse principle for certain quartic hypersurfaces’, Proc. Amer. Math. Soc. 139(12) (2011), 4293–4305.CrossRefGoogle Scholar
Reichardt, H., ‘Einige im Kleinen überall lösbare, im Grossen Unlösbare diophantische Gleichungen’, J. reine angew. Math. 184 (1942), 12–18.CrossRefGoogle Scholar
Selmer, E. S., ‘The Diophantine equation  $a{x}^3+b{y}^3+c{z}^3=0$ ’, Acta Math. 85 (1951), 203–362.CrossRefGoogle Scholar
Serre, J.-P., A Course in Arithmetic, Graduate Texts in Mathematics, 7 (Springer, New York, 1996).Google Scholar
Silverman, J. H., The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106 (Springer, New York, 1986).Google Scholar
Skorobogatov, A., ‘Beyond the Manin obstruction’, Invent. Math. 135 (1999), 399–424.CrossRefGoogle Scholar
Swinnerton-Dyer, H. P. F., ‘Arithmetic of diagonal quartic surfaces. II’, Proc. Lond. Math. Soc. (3) 80(3) (2000), 513–544.CrossRefGoogle Scholar