Given any global field of characteristic , we construct a Châtelet surface over 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 , thereby showing that the étale-Brauer obstruction is insufficient to explain all failures of the Hasse principle over a global field of any characteristic.
Soit un corps global de caractéristique . On construit une surface de Châtelet sur possédant une obstruction de Brauer-Manin au principe de Hasse. Cette construction étend un résultat de Poonen à la caractéristique , 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.
Keywords: Hasse principle, Brauer-Manin obstruction, Châtelet surface, rational points
@article{JTNB_2012__24_1_231_0, author = {Viray, Bianca}, 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}, mrnumber = {2914907}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.794/} }
TY - JOUR AU - Viray, Bianca TI - Failure of the Hasse principle for Châtelet surfaces in characteristic $2$ JO - Journal de théorie des nombres de Bordeaux PY - 2012 SP - 231 EP - 236 VL - 24 IS - 1 PB - Société Arithmétique de Bordeaux UR - http://archive.numdam.org/articles/10.5802/jtnb.794/ DO - 10.5802/jtnb.794 LA - en ID - JTNB_2012__24_1_231_0 ER -
%0 Journal Article %A Viray, Bianca %T Failure of the Hasse principle for Châtelet surfaces in characteristic $2$ %J Journal de théorie des nombres de Bordeaux %D 2012 %P 231-236 %V 24 %N 1 %I Société Arithmétique de Bordeaux %U http://archive.numdam.org/articles/10.5802/jtnb.794/ %R 10.5802/jtnb.794 %G en %F JTNB_2012__24_1_231_0
Viray, Bianca. Failure of the Hasse principle for Châtelet surfaces in characteristic $2$. Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 1, pp. 231-236. doi : 10.5802/jtnb.794. http://archive.numdam.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 | Zbl
[2] Philippe Gille, Tamás Szamuely, Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics, vol. 101, Cambridge University Press, Cambridge, 2006. | MR
[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
[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 | Zbl
[5] Bjorn Poonen, Existence of rational points on smooth projective varieties. J. Eur. Math. Soc. (JEMS) 11 (2009), no. 3, 529–543. | MR
[6] Bjorn Poonen, Insufficiency of the Brauer-Manin obstruction applied to étale covers. Ann. of Math. (2) 171 (2010), no. 3, 2157–2169. | MR
[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 | Zbl
[8] Alexei Skorobogatov, Torsors and rational points. Cambridge Tracts in Mathematics, vol. 144, Cambridge University Press, Cambridge, 2001. | MR
Cited by Sources: