Efficient evaluation of the Brauer-Manin obstruction

The Brauer-Manin obstruction is a concept which has been very effective in finding counter-examples to the Hasse principle, that is, sets of polynomial equations which have solutions in every completion of the rational numbers but have no rational solutions. The standard way of calculating the Brauer-Manin obstruction involves listing all the p-adic solutions to some accuracy, at finitely many primes p; this is a process which may be time-consuming. The result described in this paper shows that, at some primes, we do not need to list all p-adic solutions, but only those lying over a closed subset; and, at other primes, we need only to list solutions modulo p.