Tokyo Journal of Mathematics

Certain Forms Violate the Hasse Principle

Dong Quan Ngoc NGUYEN

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Abstract

A family of smooth geometrically irreducible curves violates the Hasse principle if they have local points everywhere, but they possesses no global points. In this paper, we show how to construct non-constant algebraic families of forms of degree $4k$ that violate the Hasse principle. Some examples of non-constant algebraic families of forms of degrees 12 and 24 that violate the Hasse principle are given to illustrate the method.

Article information

Source
Tokyo J. of Math. Volume 40, Number 1 (2017), 277-299.

Dates
First available in Project Euclid: 8 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1502179228

Digital Object Identifier
doi:10.3836/tjm/1502179228

Citation

NGUYEN, Dong Quan Ngoc. Certain Forms Violate the Hasse Principle. Tokyo J. of Math. 40 (2017), no. 1, 277--299. doi:10.3836/tjm/1502179228. https://projecteuclid.org/euclid.tjm/1502179228.


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References

  • M. Bhargava, Most hyperelliptic curves over $\bQ$ have no rational points, Preprint (2013). Available at http://arxiv.org/pdf/1308.0395.pdf.
  • M. Bhargava, A positive proportion of plane cubics fail the Hasse principle, Preprint (2014). Available at http://arxiv.org/pdf/1402.1131v1.pdf.
  • W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, Computational algebra and number theory (London, 1993), J. Symbolic Comput. \bf24 (1997), no. 3–4, 235–265.
  • H. Cohen, Number Theory, Volume I: Tools and Diophantine equations, Graduate Texts in Math. 239, Springer-Verlag (2007).
  • N. N. Dong Quan, On the Hasse principle for certain quartic hypersurfaces, Proc. Amer. Math. Soc. 139 (2011), no. 12, 4293–4305.
  • N. N. Dong Quan, The Hasse principle for certain hyperelliptic curves and forms, Quarterly Journal of Mathematics \bf64 (2013), no. 1, 253–268.
  • M. Fujiwara and M. Sudo, Some forms of odd degree for which the Hasse principle fails, Pacific J. Math. \bf67 (1976), no. 1, 161–169.
  • J. Jahnel, Brauer groups, Tamagawa measures, and rational points on algebraic varieties, Mathematical Surveys and Monographs, \bf198, American Mathematical Society, Providence, RI, 2014.
  • B. Poonen, An explicit family of genus-one curves violating the Hasse principle, J. Théor. Nombres Bordeaux \bf13 (2001), no. 1, 263–274.
  • E. S. Selmer, The Diophantine equation $ax^3 + by^3 + cz^3 = 0$, Acta Math. \bf85 (1951), 203–362.