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Pathologies of the Brauer-Manin obstruction. (English) Zbl 1365.14025

Let \(k\) be a number field and let \(X\) be a proper smooth \(k\)-variety. In the 1970s, Manin defined a pairing between the set of adèlic points \(X(\mathbb{A}_{k})\) and the Brauer group \(\text{Br}(X)\) to explain the failure of the Hasse principle for rational points on \(X\): if the subset of adelic points \(X(\mathbb{A}_{k})^{\text{Br}}\) that are orthogonal to the Brauer group is empty then there are no \(k\)-rational points on \(X\). In [Ann. Math. (2) 171, No. 3, 2157–2169 (2010; Zbl 1284.11096)], B. Poonen constructed 3-dimensional examples \(X\) such that the failure of Hasse principle can not be explained by such an obstruction even applied to étale covers. They are fibrations in Châtelet surfaces over curves that possess only finitely many \(k\)-rational points, over which the fibres violate the Hasse principle. A similar idea was also used to produce 2-dimensional examples by Y. Harpaz and A. N. Skorobogatov [Ann. Sci. Éc. Norm. Supér. (4) 47, No. 4, 765–778 (2014; Zbl 1308.14024)].
In the paper under review, the authors produce more examples:
a conic bundle surface \(X\to E\) over a real quadratic field \(k\), where \(E\) is an elliptic curve such that \(E(k)=\{0\}\);
a 3-fold over an arbitrary real number field \(k\), which is a family \(X\to C\) of 2-dimensional quadrics parameterised by a curve \(C\) with exactly one \(k\)-point;
a 3-fold over an arbitrary number field \(k\), which is a family \(X\to C\) of geometrically rational surfaces parameterised by a curve with exactly one \(k\)-point, the fiber above which is singular.
The main argument is similar to Poonen’s. The new idea is that to guarantee the existence of local points orthogonal to the Brauer group, they make use of either a deformation for a real place or a deformation along a rational curve defined over a non-archimedean completion of \(k\).
Furthermore, in [Am. J. Math. 139, No. 2, 417–431 (2017; Zbl 1360.14068)], A. Smeets has constructed examples with trivial Albanese variety (even simply connected examples if the abc conjecture is true).

MSC:

14F22 Brauer groups of schemes
11G35 Varieties over global fields
14G05 Rational points
14G25 Global ground fields in algebraic geometry

Software:

SageMath
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References:

[1] Bashmakov, M.I.: The cohomology of abelian varieties over a number field. Uspekhi Mat. Nauk 27(6), 25-66 (1972) (Russian). English translation: Russ. Math. Surv. 27:6, 25-70 (1972) · Zbl 0256.14016
[2] Basile, C.L., Skorobogatov, A.N.: On the Hasse principle for bielliptic surfaces. In: Number Theory and Algebraic Geometry. London Math. Soc. Lecture Note Ser. 303, pp. 31-40. Cambridge University Press (2003) · Zbl 1073.14052
[3] Colliot-Thélène, J.-L.: Zéro-cycles de degré 1 sur les solides de Poonen. Bull. Soc. Math. Fr. 138, 249-257 (2010) · Zbl 1205.11075
[4] Colliot-Thélène J.-L., Sansuc, J.-J.: La descente sur les variétés rationnelles. Journées de géométrie algébrique d’Angers (Juillet 1979), A. Beauville, éd. 223-237. Sijthof and Noordhof (1980) · Zbl 0451.14018
[5] Colliot-Thélène, J.-L., Coray, D., Sansuc, J.-J.: Descente et principe de Hasse pour certaines variétés rationnelles. J. Reine Angew. Math. 320, 150-191 (1980) · Zbl 0434.14019
[6] Colliot-Thélène, J.-L., Swinnerton-Dyer, S.P.: Hasse principle and weak approximation for pencils of Severi-Brauer and similar varieties. J. Reine Angew. Math. 453, 49-112 (1994) · Zbl 0805.14010
[7] Dokchitser, V., Dokchitser, T.: Surjectivity of mod \[2^n2\] n representations of elliptic curves. Math. Z. 272, 961-964 (2012) · Zbl 1315.11046 · doi:10.1007/s00209-011-0967-7
[8] Elkies, N.: Elliptic curves with 3-adic Galois representation surjective mod 3 but not mod 9. arXiv:math/0612734 · Zbl 1204.11088
[9] Frossard, E.: Obstruction de Brauer-Manin pour les zéro-cycles sur des fibrations en variétés de Severi-Brauer. J. Reine Angew. Math. 557, 81-101 (2003) · Zbl 1098.14015
[10] Grothendieck, A.: Le groupe de Brauer. In: Dix Exposés sur la Cohomologie des Schémas. North-Holland and Masson, pp. 46-188 (1968) · Zbl 0198.25901
[11] Harpaz, Y., Skorobogatov, A.N.: Singular curves and the étale Brauer-Manin obstruction for surfaces. Ann. Sci. École Norm. Sup. 47, 765-778 (2014) · Zbl 1308.14024
[12] Hartshorne, R.: Algebraic Geometry, Graduate Texts in Mathematics, vol. 52. Springer, Berlin (1977) · Zbl 0367.14001 · doi:10.1007/978-1-4757-3849-0
[13] Iskovskikh, V.A.: A counterexample to the Hasse principle for a system of two quadratic forms in five variables. Mat. Zametki 10, 253-257 (1971) (Russian). English translation: Math. Notes 10, 575-577 (1971) · Zbl 0221.10028
[14] Jones, N.: Almost all elliptic curves are Serre curves. Trans. Am. Math. Soc. 362, 1547-1570 (2010) · Zbl 1204.11088 · doi:10.1090/S0002-9947-09-04804-1
[15] Knus, M.-A., Merkurjev, A., Rost, M., Tignol, J.-P.: The Book of Involutions, vol. 44. AMS Colloquium Publications, New York (1998)
[16] Manin, Y.I.: Rational surfaces over perfect fields. Publ. Math. IHÉS 30, 55-97 (1966) (Russian); 99-113 (English summary) · Zbl 0171.41701
[17] Manin, Y.I.: Le groupe de Brauer-Grothendieck en géométrie diophantienne. In: Actes du Congrès International des Mathématiciens (Nice, 1970), Tome, vol. 1, pp. 401-411. Gauthier-Villars (1971) · Zbl 1098.14015
[18] Mazur, B., Rubin, K.: Ranks of twists of elliptic curves and Hilbert’s tenth problem. Invent. Math. 181, 541-575 (2010) · Zbl 1227.11075 · doi:10.1007/s00222-010-0252-0
[19] Poonen, B.: Insufficiency of the Brauer-Manin obstruction applied to étale covers. Ann. Math. 171, 2157-2169 (2010) · Zbl 1284.11096 · doi:10.4007/annals.2010.171.2157
[20] Poonen, B.: Curves over every global field violating the local-global principle. Zap. Nauchn. Sem. POMI 377 (2010), Issledovaniya po Teorii Chisel 10, 141-147, 243-244. Reprinted in: J. Math. Sci. (N.Y.) 171, 782-785 (2010) · Zbl 1294.11108
[21] Sarnak, P., Wang, L.: Some hypersurfaces in \[{\mathbb{P}}^4\] P4 and the Hasse-principle. C. R. Acad. Sci. Paris Sér. I Math. 321, 319-322 (1995) · Zbl 0857.14013
[22] Serre, J.-P.: Sur les groupes de congruence des variétés abéliennes, I. Izv. Akad. Nauk SSSR 28, 3-18 (1964) · Zbl 0128.15601
[23] Serre, J.-P.: Sur les groupes de congruence des variétés abéliennes, II. Izv. Akad. Nauk SSSR 35, 731-737 (1971) · Zbl 0222.14025
[24] Serre, J.-P.: Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15, 259-331 (1972) · Zbl 0235.14012 · doi:10.1007/BF01405086
[25] SGA1: Revêtements étales et groupe fondamental. In: Séminaire de géométrie algébrique du Bois Marie 1960-61 dirigé par A. Grothendieck. Documents Mathématiques 3. Soc. Math. France, Paris (2003) · Zbl 1039.14001
[26] Skorobogatov, A.N.: Arithmetic on certain quadric bundles of relative dimension 2. I. J. Reine Angew. Math. 407, 57-74 (1990) · Zbl 0692.14001
[27] Skorobogatov, A.N.: Beyond the Manin obstruction. Invent. Math. 135, 399-424 (1999) · Zbl 0951.14013 · doi:10.1007/s002220050291
[28] Skorobogatov, A.: Torsors and Rational Points. Cambridge University Press, Cambridge (2001) · Zbl 0972.14015 · doi:10.1017/CBO9780511549588
[29] Skorobogatov, A.N.: Descent obstruction is equivalent to étale Brauer-Manin obstruction. Math. Ann. 344, 501-510 (2009) · Zbl 1180.14017 · doi:10.1007/s00208-008-0314-4
[30] Stein, W.A., et al.: Sage Mathematics Software. http://www.sagemath.org · Zbl 0222.14025
[31] Wang, L.: Brauer-Manin obstruction to weak approximation on abelian varieties. Isr. J. Math. 94, 189-200 (1996) · Zbl 0870.14032 · doi:10.1007/BF02762704
[32] Wittenberg, O.: Zéro-cycles sur les fibrations au-dessus d’une courbe de genre quelconque. Duke Math. J. 161, 2113-2166 (2012) · Zbl 1248.14030 · doi:10.1215/00127094-1699441
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