Failure of the Hasse principle for Châtelet surfaces in characteristic 2
Journal de Théorie des Nombres de Bordeaux, Tome 24 (2012) no. 1, pp. 231-236.

Soit k un corps global de caractéristique 2. On construit une surface de Châtelet sur k possédant une obstruction de Brauer-Manin au principe de Hasse. Cette construction étend un résultat de Poonen à la caractéristique 2, en montrant ainsi que l’obstruction de Brauer-Manin après revêtement fini étale n’est pas suffisante pour expliquer tous les contre-exemples au principe de Hasse sur un corps global arbitraire.

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.

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DOI : https://doi.org/10.5802/jtnb.794
Classification : 11G35,  14G05,  14G25,  14G40
Mots clés : Hasse principle, Brauer-Manin obstruction, Châtelet surface, rational points
@article{JTNB_2012__24_1_231_0,
     author = {Bianca Viray},
     title = {Failure of the {Hasse} principle for {Ch\^atelet} surfaces in characteristic $2$},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {231--236},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {24},
     number = {1},
     year = {2012},
     doi = {10.5802/jtnb.794},
     zbl = {pre06075028},
     mrnumber = {2914907},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.794/}
}
Bianca Viray. Failure of the Hasse principle for Châtelet surfaces in characteristic $2$. Journal de Théorie des Nombres de Bordeaux, Tome 24 (2012) no. 1, pp. 231-236. doi : 10.5802/jtnb.794. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.794/

[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