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The reciprocity obstruction for rational points on compactifications of torsors under tori over fields with global duality. (English) Zbl 1025.14003
Summary: This paper studies the reciprocity obstruction to the local-global principle for compactifications of torsors under tori over a generalised global field of characteristic zero. Under a non-divisibility condition on the second Tate-Shafarevich group for tori, it is shown that the reciprocity obstruction is the only obstruction to the local-global principle. This gives in particular an elegant unified proof of J.-J. Sansuc’s result [J. Reine Angew. Math. 327, 12-80 (1981; Zbl 0468.14007)] on the Brauer-Manin obstruction for compactifications of torsors under tori over number fields, and Scheiderer’s result on the reciprocity obstruction for compactifications of torsors under tori over $$p$$-adic function fields [see C. Scheiderer and J. van Hamel, Math. Ann. 326, 155-183 (2003; Zbl 1050.14016)].
##### MSC:
 14G25 Global ground fields in algebraic geometry 14M17 Homogeneous spaces and generalizations 14F20 Étale and other Grothendieck topologies and (co)homologies 11G35 Varieties over global fields 14G40 Arithmetic varieties and schemes; Arakelov theory; heights
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