The reciprocity obstruction for rational points on compactifications of torsors under tori over fields with global duality

Abstract

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.