The Reciprocity Obstruction for Rational Points on Compactifications of Torsors under Tori over Fields with Global Duality

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 Sansuc's result 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.

2000 Mathematics Subject Classification: 14G25 11G35 14F20 11E72

Keywords and Phrases: Arithmetic algebraic geometry, Diophantine geometry, Varieties over global fields, Brauer--Manin obstruction, Homogeneous spaces over global fields

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