no. 2
Order 5 Brauer–Manin obstructions to the integral Hasse principle on log K3 surfaces
Lyczak, Julian1
1 IST Austria Am Campus 1 3400 Klosterneuburg (Austria)
Annales de l'Institut Fourier, Volume 73 (2023) no. 2, pp. 447-478.

We construct families of log K3 surfaces and study the arithmetic of their members. We use this to produce explicit surfaces with an order 5 Brauer–Manin obstruction to the integral Hasse principle.

Supplementary Materials:
Supplementary materials for this article are supplied as separate files:

Nous construisons plusieurs familles de surfaces log K3 et en étudions l’arithmétique. Nous en déduisons des exemples explicites de surfaces avec une obstruction de Brauer–Manin d’ordre 5 au principe de Hasse entier.

Compléments :
Des compléments sont fournis pour cet article dans les fichiers suivants :

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3529
Classification: 14G12, 14J26, 14F22, 11G35
Keywords: Integral points, log K3 surface, integral Hasse principle, Brauer–Manin obstruction
Mot clés : Points entiers, surface log K3, principe de Hasse entier, obstruction de Brauer–Manin
Lyczak, Julian 1

1 IST Austria Am Campus 1 3400 Klosterneuburg (Austria)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{AIF_2023__73_2_447_0,
     author = {Lyczak, Julian},
     title = {Order $5$ {Brauer{\textendash}Manin} obstructions to the integral {Hasse} principle on log {K3} surfaces},
     journal = {Annales de l'Institut Fourier},
     pages = {447--478},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {73},
     number = {2},
     year = {2023},
     doi = {10.5802/aif.3529},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3529/}
}
TY  - JOUR
AU  - Lyczak, Julian
TI  - Order $5$ Brauer–Manin obstructions to the integral Hasse principle on log K3 surfaces
JO  - Annales de l'Institut Fourier
PY  - 2023
SP  - 447
EP  - 478
VL  - 73
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3529/
DO  - 10.5802/aif.3529
LA  - en
ID  - AIF_2023__73_2_447_0
ER  - 
%0 Journal Article
%A Lyczak, Julian
%T Order $5$ Brauer–Manin obstructions to the integral Hasse principle on log K3 surfaces
%J Annales de l'Institut Fourier
%D 2023
%P 447-478
%V 73
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3529/
%R 10.5802/aif.3529
%G en
%F AIF_2023__73_2_447_0
Lyczak, Julian. Order $5$ Brauer–Manin obstructions to the integral Hasse principle on log K3 surfaces. Annales de l'Institut Fourier, Volume 73 (2023) no. 2, pp. 447-478. doi : 10.5802/aif.3529. https://aif.centre-mersenne.org/articles/10.5802/aif.3529/

[1] Adams, William W.; Loustaunau, Philippe An introduction to Gröbner bases, Graduate Studies in Mathematics, 3, American Mathematical Society, Providence, RI, 1994, xiv+289 pages | DOI | MR | Zbl

[2] Berg, Jennifer; Várilly-Alvarado, Anthony Odd order obstructions to the Hasse principle on general K3 surfaces, Math. Comp., Volume 89 (2020) no. 323, pp. 1395-1416 | DOI | MR | Zbl

[3] Bright, Martin Efficient evaluation of the Brauer–Manin obstruction, Math. Proc. Cambridge Philos. Soc., Volume 142 (2007) no. 1, pp. 13-23 | DOI | MR | Zbl

[4] Bright, Martin; Loughran, Daniel Brauer-Manin obstruction for Erdös–Straus surfaces (2019) (https://arxiv.org/abs/1908.02526)

[5] Bright, Martin; Lyczak, Julian A uniform bound on the Brauer groups of certain log K3 surfaces, Michigan Math. J., Volume 68 (2019) no. 2, pp. 377-384 | DOI | MR | Zbl

[6] Bright, Martin James Computations on diagonal quartic surfaces, Ph. D. Thesis, University of Cambridge (2002)

[7] Colliot-Thélène, J.-L.; Wei, Dasheng; Xu, Fei Brauer–Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms, Compos. Math., Volume 145 (2009) no. 2, pp. 309-363 | DOI | MR | Zbl

[8] Colliot-Thélène, J.-L.; Wei, Dasheng; Xu, Fei Brauer-Manin obstruction for Markoff surfaces (2018) (https://arxiv.org/abs/1808.01584)

[9] Colliot-Thélène, J.-L.; Wittenberg, O. Groupe de Brauer et points entiers de deux familles de surfaces cubiques affines, Amer. J. Math., Volume 134 (2012) no. 5, pp. 1303-1327 | DOI | MR | Zbl

[10] Corn, Patrick; Nakahara, Masahiro Brauer–Manin obstructions on degree 2 K3 surfaces, Res. Number Theory, Volume 4 (2018) no. 3, 33, 16 pages | DOI | MR | Zbl

[11] Demazure, Michel Surfaces de Del Pezzo: II–V, Séminaire sur les Singularités des Surfaces (Lecture Notes in Mathematics), Volume 777, Springer, Berlin, 1980, pp. 21-69 | DOI | Zbl

[12] Enriques, Federigo Sulle irrazionalità da cui può farsi dipendere la risoluzione d’un’ equazione algebricaf (xyz)=0 con funzioni razionali di due parametri, Math. Ann., Volume 49 (1897) no. 1, pp. 1-23 | DOI | Zbl

[13] Ghosh, Amit; Sarnak, Peter Integral points on Markoff type cubic surfaces, Inventiones mathematicae (2022), pp. 1-61 | MR | Zbl

[14] González-Sánchez, Jon; Harrison, Michael; Polo-Blanco, Irene; Schicho, Josef Algorithms for Del Pezzo surfaces of degree 5 (construction, parametrization), J. Symbolic Comput., Volume 47 (2012) no. 3, pp. 342-353 | DOI | MR | Zbl

[15] Grothendieck, Alexandre Le groupe de Bauer III: exemples et compléments, IHES, 1966

[16] Harpaz, Yonatan Geometry and arithmetic of certain log K3 surfaces, Ann. Inst. Fourier (Grenoble), Volume 67 (2017) no. 5, pp. 2167-2200 | DOI | Numdam | MR | Zbl

[17] Hartshorne, Robin Algebraic geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, New York-Heidelberg, 1977, xvi+496 pages | DOI | MR | Zbl

[18] Ieronymou, Evis; Skorobogatov, Alexei N. Odd order Brauer–Manin obstruction on diagonal quartic surfaces, Adv. Math., Volume 270 (2015), pp. 181-205 | DOI | MR | Zbl

[19] Jahnel, Jörg; Schindler, Damaris On integral points on degree four del Pezzo surfaces, Israel J. Math., Volume 222 (2017) no. 1, pp. 21-62 | DOI | MR | Zbl

[20] de Jong, Aise Johan A result of Gabber, Volume 25, 2003, pp. 36-57 (available at http://www.math.columbia.edu/~dejong/papers/2-gabber.pdf)

[21] Loughran, Daniel; Mitankin, Vladimir Integral Hasse principle and strong approximation for Markoff surfaces, Int. Math. Res. Not. IMRN (2021) no. 18, pp. 14086-14122 | DOI | MR | Zbl

[22] Lyczak, JT Arithmetic of affine del Pezzo surfaces, Ph. D. Thesis, Leiden (2019)

[23] Manin, Yu I. Le groupe de Brauer–Grothendieck en géométrie diophantienne, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1 (1971), pp. 401-411 | MR | Zbl

[24] Manin, Yu I Cubic forms. Algebra, geometry, arithmetic, North-Holland Math. Libr., 4, Elsevier (North-Holland), Amsterdam, 1986 | Zbl

[25] Poonen, Bjorn Rational points on varieties, Graduate Studies in Mathematics, 186, American Mathematical Society, Providence, RI, 2017, xv+337 pages | DOI | MR | Zbl

[26] Skorobogatov, Alexei N. On a theorem of Enriques–Swinnerton–Dyer, Ann. Fac. Sci. Toulouse Math. (6), Volume 2 (1993) no. 3, pp. 429-440 | DOI | Numdam | MR | Zbl

[27] Skorobogatov, Alexei N. Diagonal quartic surfaces, Oberwolfach Rep., Volume 33 (2009), pp. 76-79

[28] Skorobogatov, Alexei N.; Zarhin, Yuri G. Kummer varieties and their Brauer groups (2016) (https://arxiv.org/abs/1612.05993)

[29] Swinnerton-Dyer, H. P. F. Rational points on del Pezzo surfaces of degree 5, Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer School in Math.) (1972), pp. 287-290 | MR | Zbl

[30] Várilly-Alvarado, Anthony Arithmetic of del Pezzo surfaces, Birational geometry, rational curves, and arithmetic (Simons Symp.), Springer, Cham, 2013, pp. 293-319 | DOI | MR | Zbl

[31] Weibel, Charles A. An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, Cambridge, 1994, xiv+450 pages | DOI | MR | Zbl

Cited by Sources: