We construct del Pezzo surfaces of degree violating the Hasse principle explained by the Brauer-Manin obstruction. Using these del Pezzo surfaces, we show that there are algebraic families of surfaces violating the Hasse principle explained by the Brauer-Manin obstruction. Various examples are given.
Nous construisons des surfaces de del Pezzo de degré violant le principe de Hasse expliqué par l’obstruction de Brauer-Manin. En utilisant ces surfaces de del Pezzo de degré , nous montrons qu’il y a des familles algébriques de surfaces violant le principe de Hasse expliqué par l’obstruction de Brauer-Manin. Divers exemples sont donnés.
@article{JTNB_2012__24_2_447_0,
author = {Dong Quan Ngoc Nguyen},
title = {The arithmetic of certain del {Pezzo} surfaces and {K3} surfaces},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {447--460},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {24},
number = {2},
year = {2012},
doi = {10.5802/jtnb.805},
mrnumber = {2950701},
zbl = {1268.14020},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.805/}
}
TY - JOUR AU - Dong Quan Ngoc Nguyen TI - The arithmetic of certain del Pezzo surfaces and K3 surfaces JO - Journal de théorie des nombres de Bordeaux PY - 2012 SP - 447 EP - 460 VL - 24 IS - 2 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.805/ DO - 10.5802/jtnb.805 LA - en ID - JTNB_2012__24_2_447_0 ER -
%0 Journal Article %A Dong Quan Ngoc Nguyen %T The arithmetic of certain del Pezzo surfaces and K3 surfaces %J Journal de théorie des nombres de Bordeaux %D 2012 %P 447-460 %V 24 %N 2 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.805/ %R 10.5802/jtnb.805 %G en %F JTNB_2012__24_2_447_0
Dong Quan Ngoc Nguyen. The arithmetic of certain del Pezzo surfaces and K3 surfaces. Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 2, pp. 447-460. doi : 10.5802/jtnb.805. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.805/
[1] B.J. Birch and H.P.F. Swinnerton-Dyer, The Hasse problem for rational surfaces. J. Reine Angew. Math. 274/275 (1975), 164–174, Collection of articles dedicated to Helmut Hasse on his seventy-fifth birthday, III. | MR | Zbl
[2] V. Bouniakowsky, Nouveaux théorèms à la distinction des nombres premiers et à la décomposition des entiers en facteurs. Sc. Math. Phys. 6 (1857), pp. 305–329.
[3] H. Cohen, Number Theory, Volume I: Tools and Diophantine equations. Graduate Texts in Math. 239, Springer-Verlag, 2007. | MR
[4] J-L. Colliot-Thélène and B. Poonen, Algebraic families of nonzero elements of Shafarevich-Tate groups. J. Amer. Math. Soc. 13 (2000), no. 1, 83–99. | MR
[5] P. Corn, Del Pezzo surfaces and the Brauer-Manin obstruction. Ph.D. thesis, University of California, Berkeley, 2005. | MR
[6] D. F. Coray and C. Manoil, On large Picard groups and the Hasse principle for curves and surfaces. Acta. Arith. 76 (1996), pp. 165–189. | MR | Zbl
[7] W. Hürlimann, Brauer group and Diophantine geometry: A cohomological approach, in: Brauer Groups in Ring Theory and Algebraic Geometry, Wilrijk, 1981. In: Lecture Notes in Math., vol. 917, Springer, Berlin, 1982, pp. 43–65. | MR | Zbl
[8] V.A. Iskovskikh, A counterexample to the Hasse principle for systems of two quadratic forms in five variables. Mat. Zametki 10 (1971), 253–257. | MR | Zbl
[9] C.E. Lind, Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins. Thesis, University of Uppasala, 1940. | MR | Zbl
[10] Yu.I. Manin, Le groupe de Brauer-Grothendieck en géométrie diophantienne. In Actes du Congrès International des Mathématiciens, (Nice, 1970), pp. 401–411. | MR | Zbl
[11] B. Poonen, An explicit family of genus-one curves violating the Hasse principle. J. Théor. Nombres Bordeaux 13 (2001), no. 1, 263–274. | Numdam | MR
[12] B. Poonen, Existence of rational points on smooth projective varieties. J. Eur. Math. Soc. 11 (2009), no. 3, pp. 529–543. | MR
[13] H. Reichardt, Einige im Kleinen überall lösbare, im Grossen unlösbare diophantische Gleichungen. J. Reine Angew. Math. 184 (1942), pp. 12–18. | MR
[14] M. Reid, The complete intersection of two or more quadrics. Ph.D thesis, University of Cambridge, 1972.
[15] A.N. Skorobogatov, Torsors and rational points, CTM 144. Cambridge Univ. Press, 2001. | MR
[16] H.P.F. Swinnerton-Dyer, Brauer-Manin obstructions on some Del Pezzo surfaces. Math. Proc. Cam. Phil. Soc. 125 (1999), pp. 193–198. | MR | Zbl
[17] Oliver Wittenberg, Intersections de deux quadriques et pinceaux de courbes de genre . Lecture Notes in Mathematics. 1901, Springer, Berlin, 2007.
Cited by Sources: