The reciprocity obstruction for rational points on compactifications of torsors under tori over fields with global duality
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 \textit{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 \textit{C. Scheiderer} and \textit{J. van Hamel}, Math. Ann. 326, 155-183 (2003; Zbl 1050.14016)].
obstruction to the local-global principle
scientific article; zbMATH DE number 2000815