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The Brauer–Manin obstruction for integral points on curves

Published online by Cambridge University Press:  24 June 2010

DAVID HARARI  and
JOSÉ FELIPE VOLOCH
DAVID HARARI
Affiliation:
Université Paris-Sud, Laboratoire de Mathématiques d'Orsay, Orsay Cedex, F-91405, France. e-mail: [email protected]
JOSÉ FELIPE VOLOCH
Affiliation:
Department of Mathematics, University of Texas, Austin, TX 78712, U.S.A. e-mail: [email protected]
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Abstract

We discuss the question of whether the Brauer–Manin obstruction is the only obstruction to the Hasse principle for integral points on affine hyperbolic curves. In the case of rational curves we conjecture a positive answer, we prove that this conjecture can be given several equivalent formulations and we relate it to an old conjecture of Skolem. Finally, we show that for elliptic curves minus one point a strong version of the question (describing the set of integral points by local conditions) has a negative answer.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 149 , Issue 3 , November 2010 , pp. 413 - 421
Copyright
Copyright © Cambridge Philosophical Society 2010

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References

REFERENCES

[1]Bruin, N. and Stoll, M.Deciding existence of rational points on curves: an experiment. Experiment. Math. 17 (2008), no. 2, 181–189.Google Scholar
[2]Colliot–Thélène, J. L. and Xu, FeiBrauer–Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms. Compositio Math. 145 (2009), no. 2, 309–363.Google Scholar
[3]Fossum, R. and Iversen, B.On Picard groups of algebraic fibre spaces. J. Pure Appl. Algebra 3 (1973), 269–280.CrossRefGoogle Scholar
[4]Gebel, J., Pethö, A. and Zimmer, H. G.On Mordell's equation. Compositio Math. 110 (1998), no. 3, 335–367.Google Scholar
[5]Gille, P. and Szamuely, T.Central Simple Algebras and Galois Cohomology. Cambridge Stud. Adv. Math. 101, (Cambridge University Press, Cambridge, 2006).Google Scholar
[6]Harari, D.Le défaut d'approximation forte pour les groupes algébriques commutatifs. Algebra and Number Theory 2 (2008), no. 5, 595–611.Google Scholar
[7]Harari, D. and Szamuely, T.Local-global principles for 1-motives. Duke Math. J. 143 (2008), no. 3, 531–557.Google Scholar
[8]Lang, S.Algebraic groups over finite fields. Amer. J. Math. 78 (1956), 555–563.Google Scholar
[9]Milne, J. S.Étale cohomology. Princeton Mathematical Series 33. (Princeton University Press, 1980).Google Scholar
[10]Milne, J. S.Arithmetic duality theorems (2nd edition). (BookSurge, LLC, Charleston, SC, 2006).Google Scholar
[11]Poonen, B.Heuristics for the Brauer–Manin obstruction for curves. Experiment. Math. 15 (2006), no. 4, 415–420.Google Scholar
[12]Poonen, B. and Voloch, J. F. The Brauer–Manin obstruction for subvarieties of abelian varieties over function fields. Ann. of Math. To appear.Google Scholar
[13]Ramachandran, N.Duality of Albanese and Picard 1-motives. K-Theory 22 (2001), no. 3, 271–301.Google Scholar
[14]Scharaschkin, V. Local-global problems and the Brauer–Manin obstruction (Ph.D. thesis, University of Michigan, 1999).Google Scholar
[15]Skolem, T.Anwendung exponentieller Kongruenzen zum Beweis der Unlösbarkeit gewisser diophantischer Gleichungen. Avh. Norske Vid. Akad. Oslo 12 (1937), 1–16.Google Scholar
[16]Skorobogatov, A. N.Torsors and rational points. Cambridge Tracts in Mathematics, 144 (Cambridge University Press, 2001).Google Scholar
[17]Stoll, M.Finite descent obstructions and rational points on curves. Algebra and Number Theory 2 (2008), no. 5, 595–611.Google Scholar
[18]Sun, C.-L. Adelic points of subvarieties of semi-abelian varieties over a global field of positive characteristic. In preparation.Google Scholar