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Brauer-Manin obstruction for Markoff surfaces. (English) Zbl 1478.11091

A few years ago the first and third authors [Compos. Math. 145, No. 2, 309–363 (2009; Zbl 1190.11036)] discussed obstructions to the integral Hasse principle. In the same spirit, the authors of the present work investigate in detail the Brauer groups of the affine schemes \[ \mathcal{U}_{m}: x^{2}+y^{2}+z^{2}-xyz=m,\; m\in\mathbb{Z}, \] over \(\mathbb{Z}\) and the Brauer-Manin obstruction to the integral Hasse principle for those schemes. They prove that the scheme \(\mathcal{U}_{m}\) never satisfies the strong approximation; to be precise, let \(m\in\mathbb{Z}\), let \(\mathcal{P}\) stand for set of all the primes of \(\mathbb{Q}\) including \(\infty\), let \[ A_{\mathbb{Z}}:=\prod_{p\in\mathcal{P}}\mathbb{Z}_{p}, \] suppose that \(\mathcal{U}_{m}(A_{\mathbb{Z}})\neq\emptyset\), and let \(S\) be a finite subset of \(\mathcal{P}\), then the image of the natural imbedding \(\mathcal{U}_{m}(\mathbb{Z})\hookrightarrow \mathcal{U}_{m}(A_{\mathbb{Z}})\) is not dense. As a consequence of this result, the authors exhibit infinitely many \(\log K3\) surfaces whose integral points are Zariski dense but not dense in the integral Brauer-Manin sets. Further, let \(\mathcal{U}_{m}(A_{\mathbb{Z}})^{Br}\) be the integral Brauer-Manin set of the scheme \(\mathcal{U}_{m}\) and consider the set \[ B:=\{\mathcal{U}_{m}\mid\mathcal{U}_{m}(\mathbb{Z})=\emptyset,\; \mathcal{U}_{m}(A_{\mathbb{Z}})^{Br}\neq\emptyset\} \] of the Markoff schemes for which the failure to satisfy the integral Hasse principle is not accounted by the Brauer-Manin obstruction; the authors obtain the lower estimate \[ \mathcal{N}(x)\gg\left( \frac{x}{\log x}\right)^{1/2}\; \text{ as } x\rightarrow\infty \] for the number \[ \mathcal{N}(x):=\operatorname{card}\{m\mid m\in\mathbb{Z},\; |m|<x,\; \mathcal{U}_{m}\in B\} \] of such schemes.
Reviewer: B. Z. Moroz (Bonn)

MSC:

11G35 Varieties over global fields
11D25 Cubic and quartic Diophantine equations
14F22 Brauer groups of schemes

Citations:

Zbl 1190.11036
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References:

[1] TOM M. APOSTOL, “Introduction to Analytic Number Theory”, Springer-Verlag, 1976. · Zbl 0335.10001
[2] S. BLOCH and A. OGUS, Gersten’s conjecture and the homology of schemes, Ann. Sci. Ecole Norm. Supér. (4) 7 (1974), 181-202. · Zbl 0307.14008
[3] J. BOURGAIN, A. GAMBURD and P. SARNAK, Markoff triples and strong approximation, C. R. Math. Acad. Sci. Paris 354 (2016), 131-135. · Zbl 1378.11043
[4] J.-L. COLLIOT-THÉLÈNE, Birational invariants, purity and the Gersten conjecture, In: “K-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Al-gebras”, AMS Summer Research Institute, Santa Barbara 1992, ed. W. Jacob and A. Rosen-berg, Proceedings of Symposia in Pure Mathematics, Vol. 58, Part I 1995, 1-64. · Zbl 0834.14009
[5] J.-L. COLLIOT-THÉLÈNE, A.N. SKOROBOGATOV and SIR PETER SWINNERTON-DYER, Double fibres and double covers: paucity of rational points, Acta Arith. 79 (1997), 113-135. · Zbl 0863.14011
[6] J.-L. COLLIOT-THÉLÈNE and F. XU, Brauer-Manin obstruction for integral points of homogeneous spaces and representation of integral quadratic forms, Compos. Math. 145 (2009), 309-363. · Zbl 1190.11036
[7] J.-L. COLLIOT-THÉLÈNE and O. WITTENBERG, Groupe de Brauer et points entiers de deux familles de surfaces cubiques affines, Amer. J. Math. 134 (2012), 1303-1327. · Zbl 1295.14020
[8] A. GROTHENDIECK, Le groupe de Brauer III, In: “Dix exposés sur la cohomologie des schémas”, Masson, North-Holland, 1968.
[9] P. GILLE and T. SZAMUELY, “Central Simple Algebras and Galois Cohomology”, Second Edition, Cambridge Studies in Advanced Mathematics, Vol. 165, Cambridge University Press, 2017. · Zbl 1373.19001
[10] A. GHOSH and P. SARNAK, Integral points on Markoff type cubic surfaces, arXiv: 1706.06712v2.
[11] Y. HARPAZ, Geometry and arithmetic of certain log K 3 surfaces, Ann. Inst. Fourier (Grenoble) 67 (2017), 2167-2200. · Zbl 1401.14125
[12] R. HARTSHORNE, “Algebraic Geometry”, GTM, Vol. 52, Springer, 1977. · Zbl 0367.14001
[13] J. JAHNEL and D. SCHINDLER, On integral points of degree four del Pezzo surfaces, Israel J. Math. 222 (2017), 21-62. · Zbl 1445.11055
[14] K. KATO, A Hasse principle for two dimensional global fields, J. Reine Angew. Math. 366 (1986), 142-181. · Zbl 0576.12012
[15] D. LOUGHRAN and V. MITANKIN, Integral Hasse principle and strong approximation for Markoff surfaces, Int. Math. Res. Not. IMRN, to appear, arXiv:1807.10223v3.
[16] J. S. MILNE, “Étale Cohomology”, Princeton Mathematical Series, Vol. 33, Princeton Uni-versity Press, 1980. · Zbl 0433.14012
[17] L. J. MORDELL, On the integer solutions of the equation x 2 + y 2 + z 2 + 2x yz = n, J. London Math. Soc. 28 (1953) 500-510. · Zbl 0051.27802
[18] J. NEUKIRCH, A. SCHMIDT and K. WINGBERG, “Cohomology of Number Fields”, Grundlehren der Math., Vol. 323, Springer, 2000. · Zbl 0948.11001
[19] P. SALBERGER and A. SKOROBOGATOV, Weak approximation for surfaces defined by two quadratic forms, Duke Math. J. 63 (1991), 517-536. · Zbl 0770.14019
[20] J.-J. SANSUC, Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math. 327 (1981), 12-80. · Zbl 0468.14007
[21] H.P.F. SWINNERTON-DYER, The birationality of cubic surfaces over a given field, Michi-gan Math. J. 17 (1970), 289-295. · Zbl 0224.14011
[22] Université Paris-Saclay CNRS, Laboratoire de mathématiques d’Orsay 91405, Orsay, France jlct@math.u-psud.fr
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