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CERTAIN GENERALIZED MORDELL CURVES OVER THE RATIONAL NUMBERS ARE POINTLESS

  • NGUYEN NGOC DONG QUAN (a1)

Abstract

A conjecture of Scharaschkin and Skorobogatov states that there is a Brauer–Manin obstruction to the existence of rational points on a smooth geometrically irreducible curve over a number field. In this paper, we verify the Scharaschkin–Skorobogatov conjecture for explicit families of generalized Mordell curves. Our approach uses standard techniques from the Brauer–Manin obstruction and the arithmetic of certain threefolds.

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[1]Bhargava, M., ‘Most hyperelliptic curves over $\mathbb{Q}$ have no rational points’, Preprint, 2013. Available at http://arxiv.org/pdf/1308.0395.pdf.
[2]Cohen, H., Number Theory, Vol. I: Tools and Diophantine Equations, Graduate Texts in Mathematics, 239 (Springer, New York, 2007).
[3]Jahnel, J., Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties, Mathematical Surveys and Monographs, 198 (American Mathematical Society, Providence, RI, 2014).
[4]Poonen, B., ‘Rational points on varieties’, 2008. Available at http://www-math.mit.edu/∼poonen/papers/Qpoints.pdf.
[5]Scharasckin, V., ‘Local–global problems and the Brauer–Manin obstruction’, PhD Thesis, University of Michigan, 1999.
[6]Schinzel, A., ‘Remarks on the paper ‘Sur certaines hypothèses concernant les nombres premiers’’, Acta Arith. 7 (1961–1962), 18.
[7]Schinzel, A. and Sierpiński, W., ‘Sur certaines hypothèses concernant les nombres premiers’, Acta Arith. 4 (1958), 185208; corrigendum Acta Arith. 5 (1958), 259.
[8]Skorobogatov, A. N., Torsors and Rational Points, Cambridge Tracts in Mathematics, 144 (Cambridge University Press, Cambridge, 2001).
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CERTAIN GENERALIZED MORDELL CURVES OVER THE RATIONAL NUMBERS ARE POINTLESS

  • NGUYEN NGOC DONG QUAN (a1)

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