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Bianca Viray
Failure of the Hasse principle for Châtelet surfaces in characteristic $2$
Journal de théorie des nombres de Bordeaux, 24 no. 1 (2012), p. 231-236, doi: 10.5802/jtnb.794
Article PDF | Reviews MR 2914907 | Zbl pre06075028
Class. Math.: 11G35, 14G05, 14G25, 14G40
Keywords: Hasse principle, Brauer-Manin obstruction, Châtelet surface, rational points

Résumé - Abstract

Given any global field $k$ of characteristic $2$, we construct a Châtelet surface over $k$ that fails to satisfy the Hasse principle. This failure is due to a Brauer-Manin obstruction. This construction extends a result of Poonen to characteristic $2$, thereby showing that the étale-Brauer obstruction is insufficient to explain all failures of the Hasse principle over a global field of any characteristic.


[1] Y. I. Manin, Le groupe de Brauer-Grothendieck en géométrie diophantienne. Actes du Congrès International des Mathématiciens (Nice, 1970), Gauthier-Villars, Paris, 1971, 401–411.  MR 427322 |  Zbl 0239.14010
[2] Philippe Gille, Tamás Szamuely, Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics, vol. 101, Cambridge University Press, Cambridge, 2006.  MR 2266528
[3] Jun-ichi Igusa, An introduction to the theory of local zeta functions. AMS/IP Studies in Advanced Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2000.  MR 1743467
[4] Jürgen Neukirch, Algebraic number theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder, Springer-Verlag, Berlin, 1999.  MR 1697859 |  Zbl 0956.11021
[5] Bjorn Poonen, Existence of rational points on smooth projective varieties. J. Eur. Math. Soc. (JEMS) 11 (2009), no. 3, 529–543.  MR 2505440
[6] Bjorn Poonen, Insufficiency of the Brauer-Manin obstruction applied to étale covers. Ann. of Math. (2) 171 (2010), no. 3, 2157–2169.  MR 2680407
[7] Jean-Pierre Serre, Local fields. Graduate Texts in Mathematics, vol. 67, Translated from the French by Marvin Jay Greenberg, Springer-Verlag, New York, 1979.  MR 554237 |  Zbl 0423.12016
[8] Alexei Skorobogatov, Torsors and rational points. Cambridge Tracts in Mathematics, vol. 144, Cambridge University Press, Cambridge, 2001.  MR 1845760