The arithmetic of certain del Pezzo surfaces and K3 surfaces
Dong Quan Ngoc Nguyen1
1Department of Mathematics The University of Arizona, 617 N. Santa Rita Ave, Tucson, Arizona 85721, USA
Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 2, pp. 447-460

We construct del Pezzo surfaces of degree 4 violating the Hasse principle explained by the Brauer-Manin obstruction. Using these del Pezzo surfaces, we show that there are algebraic families of K3 surfaces violating the Hasse principle explained by the Brauer-Manin obstruction. Various examples are given.

Nous construisons des surfaces de del Pezzo de degré 4 violant le principe de Hasse expliqué par l’obstruction de Brauer-Manin. En utilisant ces surfaces de del Pezzo de degré 4, nous montrons qu’il y a des familles algébriques de surfaces K3 violant le principe de Hasse expliqué par l’obstruction de Brauer-Manin. Divers exemples sont donnés.

Received:
Published online:
DOI: 10.5802/jtnb.805

Dong Quan Ngoc Nguyen  1

1 Department of Mathematics The University of Arizona, 617 N. Santa Rita Ave, Tucson, Arizona 85721, USA 
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
@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},
     year = {2012},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {24},
     number = {2},
     doi = {10.5802/jtnb.805},
     zbl = {1268.14020},
     mrnumber = {2950701},
     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

[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 K3 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. | MR | Numdam

[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 1. Lecture Notes in Mathematics. 1901, Springer, Berlin, 2007.

Cited by Sources: