We construct families of log K3 surfaces and study the arithmetic of their members. We use this to produce explicit surfaces with an order 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.
Revised:
Accepted:
Online First:
Keywords: Integral points, log K3 surface, integral Hasse principle, Brauer–Manin obstruction
@unpublished{AIF_0__0_0_A104_0,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},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
year = {2022},
doi = {10.5802/aif.3529},
language = {en},
note = {Online first},
}
TY - UNPB 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 - 2022 DA - 2022/// PB - Association des Annales de l’institut Fourier N1 - Online first UR - https://doi.org/10.5802/aif.3529 DO - 10.5802/aif.3529 LA - en ID - AIF_0__0_0_A104_0 ER -
%0 Unpublished Work %A Lyczak, Julian %T Order $5$ Brauer–Manin obstructions to the integral Hasse principle on log K3 surfaces %J Annales de l'Institut Fourier %D 2022 %I Association des Annales de l’institut Fourier %Z Online first %U https://doi.org/10.5802/aif.3529 %R 10.5802/aif.3529 %G en %F AIF_0__0_0_A104_0
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] An introduction to Gröbner bases, Graduate Studies in Mathematics, 3, American Mathematical Society, Providence, RI, 1994, xiv+289 pages | DOI | MR | Zbl
[2] 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] Efficient evaluation of the Brauer–Manin obstruction, Math. Proc. Cambridge Philos. Soc., Volume 142 (2007) no. 1, pp. 13-23 | DOI | MR | Zbl
[4] Brauer-Manin obstruction for Erdös–Straus surfaces (2019) (https://arxiv.org/abs/1908.02526)
[5] 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] Computations on diagonal quartic surfaces, Ph. D. Thesis, University of Cambridge (2002)
[7] 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] Brauer-Manin obstruction for Markoff surfaces (2018) (https://arxiv.org/abs/1808.01584)
[9] 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] Brauer–Manin obstructions on degree 2 K3 surfaces, Res. Number Theory, Volume 4 (2018) no. 3, 33, 16 pages | DOI | MR | Zbl
[11] 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] Sulle irrazionalità da cui può farsi dipendere la risoluzione d’un’ equazione algebricaf con funzioni razionali di due parametri, Math. Ann., Volume 49 (1897) no. 1, pp. 1-23 | DOI
[13] Integral points on Markoff type cubic surfaces, Inventiones mathematicae (2022), pp. 1-61
[14] 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] Le groupe de Bauer III: exemples et compléments, IHES, 1966
[16] Geometry and arithmetic of certain log K3 surfaces, Ann. Inst. Fourier (Grenoble), Volume 67 (2017) no. 5, pp. 2167-2200 | DOI | MR | Zbl
[17] Algebraic geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, New York-Heidelberg, 1977, xvi+496 pages | MR | Zbl
[18] Odd order Brauer–Manin obstruction on diagonal quartic surfaces, Adv. Math., Volume 270 (2015), pp. 181-205 | DOI | MR | Zbl
[19] On integral points on degree four del Pezzo surfaces, Israel J. Math., Volume 222 (2017) no. 1, pp. 21-62 | DOI | MR | Zbl
[20] A result of Gabber, Volume 25, 2003, pp. 36-57 (available at http://www.math.columbia.edu/~dejong/papers/2-gabber.pdf)
[21] Integral Hasse principle and strong approximation for Markoff surfaces, Int. Math. Res. Not. IMRN (2021) no. 18, pp. 14086-14122 | DOI | MR | Zbl
[22] Arithmetic of affine del Pezzo surfaces, Ph. D. Thesis, Leiden (2019)
[23] 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] Cubic forms. Algebra, geometry, arithmetic, North-Holland Math. Libr., 4, Elsevier (North-Holland), Amsterdam, 1986 | Zbl
[25] Rational points on varieties, Graduate Studies in Mathematics, 186, American Mathematical Society, Providence, RI, 2017, xv+337 pages | DOI | MR | Zbl
[26] On a theorem of Enriques–Swinnerton–Dyer, Ann. Fac. Sci. Toulouse Math. (6), Volume 2 (1993) no. 3, pp. 429-440 | MR | Zbl
[27] Diagonal quartic surfaces, Oberwolfach Rep., Volume 33 (2009), pp. 76-79
[28] Kummer varieties and their Brauer groups (2016) (https://arxiv.org/abs/1612.05993)
[29] Rational points on del Pezzo surfaces of degree , Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer School in Math.) (1972), pp. 287-290 | MR | Zbl
[30] Arithmetic of del Pezzo surfaces, Birational geometry, rational curves, and arithmetic (Simons Symp.), Springer, Cham, 2013, pp. 293-319 | DOI | MR | Zbl
[31] 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:
