Joost van Hamel

摘要

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.