Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-26T06:21:13.461Z Has data issue: false hasContentIssue false
  • English
  • Français

CERTAIN GENERALIZED MORDELL CURVES OVER THE RATIONAL NUMBERS ARE POINTLESS

Part of: Arithmetic problems. Diophantine geometry Curves Arithmetic algebraic geometry

Published online by Cambridge University Press:  28 October 2015

NGUYEN NGOC DONG QUAN
NGUYEN NGOC DONG QUAN*
Affiliation:
Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, USA email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Brauer–Manin obstructionrational pointsgeneralized curves

MSC classification

Primary: 14G25: Global ground fields
Secondary: 11G35: Varieties over global fields 14G05: Rational points 14H45: Special curves and curves of low genus
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 100 , Issue 1 , February 2016 , pp. 42 - 64
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Bhargava, M., ‘Most hyperelliptic curves over $\mathbb{Q}$ have no rational points’, Preprint, 2013. Available at http://arxiv.org/pdf/1308.0395.pdf.Google Scholar
Cohen, H., Number Theory, Vol. I: Tools and Diophantine Equations, Graduate Texts in Mathematics, 239 (Springer, New York, 2007).Google Scholar
Jahnel, J., Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties, Mathematical Surveys and Monographs, 198 (American Mathematical Society, Providence, RI, 2014).CrossRefGoogle Scholar
Poonen, B., ‘Rational points on varieties’, 2008. Available at http://www-math.mit.edu/∼poonen/papers/Qpoints.pdf.Google Scholar
Scharasckin, V., ‘Local–global problems and the Brauer–Manin obstruction’, PhD Thesis, University of Michigan, 1999.Google Scholar
Schinzel, A., ‘Remarks on the paper ‘Sur certaines hypothèses concernant les nombres premiers’’, Acta Arith. 7 (1961–1962), 1–8.CrossRefGoogle Scholar
Schinzel, A. and Sierpiński, W., ‘Sur certaines hypothèses concernant les nombres premiers’, Acta Arith. 4 (1958), 185–208; corrigendum Acta Arith. 5 (1958), 259.CrossRefGoogle Scholar
Skorobogatov, A. N., Torsors and Rational Points, Cambridge Tracts in Mathematics, 144 (Cambridge University Press, Cambridge, 2001).CrossRefGoogle Scholar