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The Brauer-Manin obstruction for subvarieties of abelian varieties over function fields. (English) Zbl 1294.11110
Summary: We prove that for a large class of subvarieties of abelian varieties over global function fields, the Brauer-Manin condition on adelic points cuts out exactly the rational points. This result is obtained from more general results concerning the intersection of the adelic points of a subvariety with the adelic closure of the group of rational points of the abelian variety.

##### MSC:
 11G35 Varieties over global fields 14K12 Subvarieties of abelian varieties 14G25 Global ground fields in algebraic geometry
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##### References:
 [1] D. Abramovich and J. F. Voloch, ”Toward a proof of the Mordell-Lang conjecture in characteristic $$p$$,” Internat. Math. Res. Notices, vol. 5, iss. 5, pp. 103-115, 1992. · Zbl 0787.14026 [2] N. Bourbaki, Algebra II. Chapters 4-7, New York: Springer-Verlag, 2003. · Zbl 1017.12001 [3] N. Bourbaki, Commutative Algebra, New York: Springer-Verlag, 1998. · Zbl 1101.13300 [4] A. Buium and J. F. Voloch, ”Integral points of abelian varieties over function fields of characteristic zero,” Math. Ann., vol. 297, iss. 2, pp. 303-307, 1993. · Zbl 0789.14017 [5] J. W. S. Cassels, ”Arithmetic on curves of genus $$1$$. III. The Tate-Šafarevi\vc and Selmer groups,” Proc. London Math. Soc., vol. 12, pp. 259-296, 1962. · Zbl 0106.03705 [6] J. W. S. Cassels, ”Arithmetic on curves of genus $$1$$. VII. The dual exact sequence,” J. Reine Angew. Math., vol. 216, pp. 150-158, 1964. · Zbl 0146.42304 [7] F. Delon, ”Separably closed fields,” in Model Theory and Algebraic Geometry, New York: Springer-Verlag, 1998, pp. 143-176. · Zbl 0925.03169 [8] C. D. González-Avilés and K. Tan, ”A generalization of the Cassels-Tate dual exact sequence,” Math. Res. Lett., vol. 14, pp. 295-302, 2007. · Zbl 1142.11045 [9] E. V. Flynn, ”The Hasse principle and the Brauer-Manin obstruction for curves,” Manuscripta Math., vol. 115, iss. 4, pp. 437-466, 2004. · Zbl 1069.11023 [10] M. J. Greenberg, ”Rational points in Henselian discrete valuation rings,” Inst. Hautes Études Sci. Publ. Math., vol. 31, pp. 59-64, 1966. · Zbl 0146.42201 [11] A. Grothendieck, ”Le groupe de Brauer. III. Exemples et compléments,” in Dix Exposés sur la Cohomologie des Schémas, Amsterdam: North-Holland, 1968, pp. 88-188. · Zbl 0198.25901 [12] E. Hrushovski, ”The Mordell-Lang conjecture for function fields,” J. Amer. Math. Soc., vol. 9, iss. 3, pp. 667-690, 1996. · Zbl 0864.03026 [13] S. Lang, Fundamentals of Diophantine Geometry, New York: Springer-Verlag, 1983, vol. 191. · Zbl 0528.14013 [14] S. Lang and A. Néron, ”Rational points of abelian varieties over function fields,” Amer. J. Math., vol. 81, pp. 95-118, 1959. · Zbl 0099.16103 [15] J. I. Manin, ”Rational points on algebraic curves over function fields,” Izv. Akad. Nauk SSSR Ser. Mat., vol. 27, pp. 1395-1440, 1963. · Zbl 0178.55102 [16] Y. I. Manin, ”Le groupe de Brauer-Grothendieck en géométrie diophantienne,” in Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, Paris: Gauthier-Villars, 1971, pp. 401-411. · Zbl 0239.14010 [17] J. I. Manin, ”Rational points on algebraic curves over function fields,” Izv. Akad. Nauk SSSR Ser. Mat., vol. 27, pp. 1395-1440, 1963. · Zbl 0178.55102 [18] J. S. Milne, ”Elements of order $$p$$ in the Tate-Šafarevi\vc group,” Bull. London Math. Soc., vol. 2, pp. 293-296, 1970. · Zbl 0205.50801 [19] J. S. Milne, ”Congruence subgroups of abelian varieties,” Bull. Sci. Math., vol. 96, pp. 333-338, 1972. · Zbl 0247.14003 [20] J. S. Milne, Étale Cohomology, Princeton, N.J.: Princeton Univ. Press, 1980. · Zbl 0433.14012 [21] J. S. Milne, Arithmetic Duality Theorems, Boston, MA: Academic Press, Inc., 1986. · Zbl 0613.14019 [22] B. Poonen, ”Heuristics for the Brauer-Manin obstruction for curves,” Experiment. Math., vol. 15, iss. 4, pp. 415-420, 2006. · Zbl 1173.11040 [23] V. Scharaschkin, ”Local-global problems and the Brauer-Manin obstruction,” PhD Thesis , University of Michigan, 1999. · Zbl 0938.11053 [24] J. Serre, ”Sur les groupes de congruence des variétés abéliennes,” Izv. Akad. Nauk SSSR Ser. Mat., vol. 28, pp. 3-20, 1964. · Zbl 0128.15601 [25] J. Serre, ”Sur les groupes de congruence des variétés abéliennes. II,” Izv. Akad. Nauk SSSR Ser. Mat., vol. 35, pp. 731-737, 1971. · Zbl 0222.14025 [26] J. Serre, Local Fields, New York: Springer-Verlag, 1979. · Zbl 0423.12016 [27] J. Serre, Lie Algebras and Lie Groups, Second ed., New York: Springer-Verlag, 1992. · Zbl 0742.17008 [28] A. Skorobogatov, Torsors and Rational Points, Cambridge: Cambridge Univ. Press, 2001. · Zbl 0972.14015 [29] M. Stoll, ”Finite descent and rational points on curves,” Algebra and Number Theory, vol. 1, pp. 349-391, 2007. · Zbl 1167.11024 [30] J. F. Voloch, ”Diophantine approximation on abelian varieties in characteristic $$p$$,” Amer. J. Math., vol. 117, iss. 4, pp. 1089-1095, 1995. · Zbl 0855.11029 [31] L. Wang, ”Brauer-Manin obstruction to weak approximation on abelian varieties,” Israel J. Math., vol. 94, pp. 189-200, 1996. · Zbl 0870.14032 [32] Y. G. Zarhin, ”Abelian varieties without homotheties,” Math. Res. Lett., vol. 14, iss. 1, pp. 157-164, 2007. · Zbl 1128.11031
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