Efficient evaluation of the Brauer–Manin obstruction

Abstract

AbstractThe 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 thep-adic solutions to some accuracy, at finitely many primesp; 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 allp-adic solutions, but only those lying over a closed subset; and, at other primes, we need only to list solutions modulop.