Order 5 Brauer–Manin obstructions to the integral Hasse principle on log K3 surfaces
Annales de l'Institut Fourier, Online first, 32 p.

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.

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.

Received:
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Accepted:
Online First:
DOI: 10.5802/aif.3529
Classification: 14G12,  14J26,  14F22,  11G35
Keywords: Integral points, log K3 surface, integral Hasse principle, Brauer–Manin obstruction
Lyczak, Julian 1

1 IST Austria Am Campus 1 3400 Klosterneuburg (Austria)
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Lyczak, Julian. Order $5$ Brauer–Manin obstructions to the integral Hasse principle on log K3 surfaces. Annales de l'Institut Fourier, Online first, 32 p.

[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

[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

[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

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

[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 | MR | Zbl

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

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