Semi-integral Brauer–Manin obstruction and quadric orbifold pairs

Mitankin, Vladimir; Nakahara, Masahiro; Streeter, Sam

doi:10.1090/tran/9170

Semi-integral Brauer–Manin obstruction and quadric orbifold pairs
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by Vladimir Mitankin, Masahiro Nakahara and Sam Streeter;

Trans. Amer. Math. Soc. 377 (2024), 4435-4480

DOI: https://doi.org/10.1090/tran/9170

Published electronically: April 19, 2024

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Abstract:

We study local-global principles for two notions of semi-integral points, termed Campana points and Darmon points. In particular, we develop a semi-integral version of the Brauer–Manin obstruction interpolating between Manin’s classical version for rational points and the integral version developed by Colliot-Thélène and Xu. We determine the status of local-global principles, and obstructions to them, in two families of orbifolds naturally associated to quadric hypersurfaces. Further, we establish a quantitative result measuring the failure of the semi-integral Brauer–Manin obstruction to account for its integral counterpart for affine quadrics.

References

Manjul Bhargava and Bjorn Poonen, The local-global principle for integral points on stacky curves, J. Algebraic Geom. 31 (2022), no. 4, 773–782. MR 4484553, DOI 10.1090/jag/796
Frédéric Campana, Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 3, 499–630 (English, with English and French summaries). MR 2097416, DOI 10.5802/aif.2027
Frédéric Campana, Fibres multiples sur les surfaces: aspects geométriques, hyperboliques et arithmétiques, Manuscripta Math. 117 (2005), no. 4, 429–461 (French, with English summary). MR 2163487, DOI 10.1007/s00229-005-0570-5
Frédéric Campana, Orbifoldes géométriques spéciales et classification biméromorphe des variétés kählériennes compactes, J. Inst. Math. Jussieu 10 (2011), no. 4, 809–934 (French, with English and French summaries). MR 2831280, DOI 10.1017/S1474748010000101
Frédéric Campana, Special orbifolds and birational classification: a survey, Classification of algebraic varieties, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2011, pp. 123–170. MR 2779470, DOI 10.4171/007-1/6
Frédéric Campana, Special manifolds, arithmetic and hyperbolic aspects: a short survey, Rational points, rational curves, and entire holomorphic curves on projective varieties, Contemp. Math., vol. 654, Amer. Math. Soc., Providence, RI, 2015, pp. 23–52. MR 3477539, DOI 10.1090/conm/654/13214
Yang Cao and Fei Xu, Strong approximation with Brauer-Manin obstruction for toric varieties, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 5, 1879–1908 (English, with English and French summaries). MR 3893760, DOI 10.5802/aif.3199
J.-L. Colliot-Thélène and A. N. Skorobogatov, Descent on fibrations over $\textbf {P}^1_k$ revisited, Math. Proc. Cambridge Philos. Soc. 128 (2000), no. 3, 383–393 (English, with French summary). MR 1744112, DOI 10.1017/S0305004199004077
Jean-Louis Colliot-Thélène and Alexei N. Skorobogatov, The Brauer-Grothendieck group, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 71, Springer, Cham, [2021] ©2021. MR 4304038, DOI 10.1007/978-3-030-74248-5
Jean-Louis Colliot-Thélène and Fei Xu, Brauer-Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms, Compos. Math. 145 (2009), no. 2, 309–363. With an appendix by Dasheng Wei and Xu. MR 2501421, DOI 10.1112/S0010437X0800376X
Jean-Louis Colliot-Thélène and Fei Xu, Strong approximation for the total space of certain quadratic fibrations, Acta Arith. 157 (2013), no. 2, 169–199. MR 3007293, DOI 10.4064/aa157-2-4
H. Darmon, Faltings plus epsilon, Wiles plus epsilon, and the generalized Fermat equation, C. R. Math. Rep. Acad. Sci. Canada 19 (1997), no. 1, 3–14. MR 1479291
Ajneet Dhillon and Ivan Kobyzev, $G$-theory of root stacks and equivariant $K$-theory, Ann. K-Theory 4 (2019), no. 2, 151–183. MR 3990783, DOI 10.2140/akt.2019.4.151
John Friedlander and Henryk Iwaniec, Ternary quadratic forms with rational zeros, J. Théor. Nombres Bordeaux 22 (2010), no. 1, 97–113 (English, with English and French summaries). MR 2675875, DOI 10.5802/jtnb.706
Andrew Granville and Olivier Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika 43 (1996), no. 1, 73–107. MR 1401709, DOI 10.1112/S0025579300011608
David Harari, Méthode des fibrations et obstruction de Manin, Duke Math. J. 75 (1994), no. 1, 221–260 (French). MR 1284820, DOI 10.1215/S0012-7094-94-07507-8
Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR 1182558, DOI 10.1007/978-1-4757-2189-8
Andrew Kresch and Yuri Tschinkel, Two examples of Brauer-Manin obstruction to integral points, Bull. Lond. Math. Soc. 40 (2008), no. 6, 995–1001. MR 2471948, DOI 10.1112/blms/bdn081
Qing Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002. Translated from the French by Reinie Erné; Oxford Science Publications. MR 1917232
Daniel Loughran and Vladimir Mitankin, Integral Hasse principle and strong approximation for Markoff surfaces, Int. Math. Res. Not. IMRN 18 (2021), 14086–14122. MR 4320804, DOI 10.1093/imrn/rnz114
C. Lv and H. Wu, The Brauer-Manin obstruction on algebraic stacks, Preprint, arXiv:2306.14426.
V. Mitankin, Failures of the integral Hasse principle for affine quadric surfaces, J. Lond. Math. Soc. (2) 95 (2017), no. 3, 1035–1052. MR 3664529, DOI 10.1112/jlms.12047
M. Nakahara and S. Streeter, Weak approximation and the Hilbert property for Campana points, Michigan Math. J., To appear, arXiv:2010.12555.
B. Nasserden and S. Xiao, The density of rational points on $\mathbb {P}^1$ with three stacky points, Preprint, arXiv:2011.06586v1.
B. Nasserden and S. Xiao, Heights and quantitative arithmetic on stacky curves, Preprint, arXiv:2108.04411v1.
Marta Pieropan, Arne Smeets, Sho Tanimoto, and Anthony Várilly-Alvarado, Campana points of bounded height on vector group compactifications, Proc. Lond. Math. Soc. (3) 123 (2021), no. 1, 57–101. MR 4307130, DOI 10.1112/plms.12391
Bjorn Poonen, Rational points on varieties, Graduate Studies in Mathematics, vol. 186, American Mathematical Society, Providence, RI, 2017. MR 3729254, DOI 10.1090/gsm/186
Tim Santens, Integral points on affine quadric surfaces, J. Théor. Nombres Bordeaux 34 (2022), no. 1, 141–161 (English, with English and French summaries). MR 4450612, DOI 10.5802/jtnb.119
T. Santens, The Brauer-Manin obstruction for stacky curves, arXiv:2210.17184.
Jean-Pierre Serre, Lettre à M. Tsfasman, Astérisque 198-200 (1991), 11, 351–353 (1992) (French, with English summary). Journées Arithmétiques, 1989 (Luminy, 1989). MR 1144337
A. Shute, Sums of four squareful numbers, Preprint, arXiv:2104.06966v1.
Alec Shute, On the leading constant in the Manin-type conjecture for Campana points, Acta Arith. 204 (2022), no. 4, 317–346. MR 4474771, DOI 10.4064/aa210430-1-7
Sam Streeter, Campana points and powerful values of norm forms, Math. Z. 301 (2022), no. 1, 627–664. MR 4405664, DOI 10.1007/s00209-021-02922-4
Tetsuya Uematsu, On the Brauer group of affine diagonal quadrics, J. Number Theory 163 (2016), 146–158. MR 3459565, DOI 10.1016/j.jnt.2015.11.015
Anthony Várilly-Alvarado and Bianca Viray, Arithmetic of del Pezzo surfaces of degree 4 and vertical Brauer groups, Adv. Math. 255 (2014), 153–181. MR 3167480, DOI 10.1016/j.aim.2014.01.004
Various authors, Stacks Project, https://stacks.math.columbia.edu.
Da Sheng Wei, Strong approximation for a toric variety, Acta Math. Sin. (Engl. Ser.) 37 (2021), no. 1, 95–103. MR 4204537, DOI 10.1007/s10114-021-8193-7