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FROM SEPARABLE POLYNOMIALS TO NONEXISTENCE OF RATIONAL POINTS ON CERTAIN HYPERELLIPTIC CURVES

  • NGUYEN NGOC DONG QUAN (a1)

Abstract

We give a separability criterion for the polynomials of the form

$$\begin{equation*} ax^{2n + 2} + (bx^{2m} + c)(d x^{2k} + e). \end{equation*}$$
Using this separability criterion, we prove a sufficient condition using the Brauer–Manin obstruction under which curves of the form
$$\begin{equation*} z^2 = ax^{2n + 2} + (bx^{2m} + c)(d x^{2k} + e) \end{equation*}$$
have no rational points. As an illustration, using the sufficient condition, we study the arithmetic of hyperelliptic curves of the above form and show that there are infinitely many curves of the above form that are counterexamples to the Hasse principle explained by the Brauer–Manin obstruction.

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References

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[2]Colliot-Thélène, J. -L., Coray, D. F. and Sansuc, J. -J., ‘Descente et principe de Hasse pour certaines variétés rationnalles’, J. reine angew. Math 320 (1980), 150191.
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[5]Dong Quan, N. N., ‘Algebraic families of hyperelliptic curves violating the Hasse principle’, 2013. Available at http://www.math.ubc.ca/∼dongquan/JTNB-algebraic-families.pdf.
[6]Dong Quan, N. N., ‘Nonexistence of rational points on certain varieties’, PhD Thesis, University of Arizona, 2012.
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[8]Jahnel, J., ‘Brauer groups, Tamagawa measures, and rational points on algebraic varieties’, Habilitationsschrift, Georg-August-Universität Göttingen, 2008.
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FROM SEPARABLE POLYNOMIALS TO NONEXISTENCE OF RATIONAL POINTS ON CERTAIN HYPERELLIPTIC CURVES

  • NGUYEN NGOC DONG QUAN (a1)

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