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Descent obstructions and Brauer–Manin obstruction in positive characteristic

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  01 June 2012

David Harari  and
José Felipe Voloch
David Harari
Affiliation:
Université Paris-Sud, Laboratoire de Mathématiques d’Orsay, Orsay Cedex, F-91405, France (david.harari@math.u-psud.fr)
José Felipe Voloch
Affiliation:
Department of Mathematics, University of Texas, Austin, TX 78712, USA (voloch@math.utexas.edu)
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Abstract

We prove that the Brauer–Manin obstruction is the only obstruction to the existence of integral points on affine varieties over global fields of positive characteristic $p$. More precisely, we show that the only obstructions come from étale covers of exponent $p$ or, alternatively, from flat covers coming from torsors under connected group schemes of exponent $p$.

Keywords

Artin–Schreier coveringBrauer–Manin obstructionGlobal field of positive characteristic

MSC classification

Secondary: 11G35: Varieties over global fields 14G17: Positive characteristic ground fields
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 12 , Issue 3 , July 2013 , pp. 545 - 551
Copyright
©Cambridge University Press 2012 

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References

Cartier, P., Une nouvelle opération sur les formes différentielles, CRAS 447 (1957), 426–428.Google Scholar
Colliot-Thélène, J.-L. and Sansuc, J.-J., La descente sur les variétés rationnelles II, Duke Math. J. 54 (1987), 375–492.CrossRefGoogle Scholar
Colliot-Thélène, J-L. and Wittenberg, O., Groupe de Brauer et points entiers de deux familles de surfaces cubiques affines, Amer. J. Math., in press.Google Scholar
Gonzalez-Avilés, C., Arithmetic duality theorems for 1-motives over function fields, J. Reine Angew. Math. 632 (2009), 203–231.Google Scholar
Harari, D. and Skorobogatov, A. N., Non abelian cohomology and rational points, Compositio Math. 130 (3) (2002), 241–273.Google Scholar
Harari, D. and Voloch, J. F., The Brauer–Manin obstruction for integral points on curves, Math. Proc. Cambridge Philos. Soc. 149 (2010), 413–421.CrossRefGoogle Scholar
Knus, M. A., Ojanguren, M. and Saltman, D. J., On Brauer groups in characteristic $p$, in Brauer groups, Evanston 1975, Lecture Notes in Mathematics, Volume 549. pp. 25–49. (Springer-Verlag, Berlin, Heidelberg, New York, 1976).Google Scholar
Milne, J. S., Arithmetic duality theorems, Second edition (BookSurge LLC, Charleston, SC, 2006).Google Scholar
Pheidas, T., Hilbert’s tenth problem for fields of rational functions over finite fields, Invent. Math. 103 (1991), 1–8.Google Scholar
Poonen, B., Insufficiency of the Brauer–Manin obstruction applied to étale covers, Ann. of Math. (2) 171 (3) (2010), 2157–2169.Google Scholar
Viray, B., Failure of the Hasse principle for Chatelet surfaces in characteristic 2, J. Théor. Nombres Bordeaux 24 (1) (2012), 231–236.Google Scholar
Weil, A., Adeles and algebraic groups, Progress in Mathematics, Volume 23 (Birkhäuser, Boston, MA, 1982).Google Scholar