On the Vanishing of the Brauer-Manin Obstruction for Normic Bundles

Abstract

We study the behaviour of the Brauer--Manin obstruction to the existence of rational points under finite field extensions. For $(p,mp)$-normic bundles over number fields, we prove that the Brauer--Manin obstruction vanishes after base change to finite extensions whose degrees satisfy suitable $p$-divisibility conditions depending on $m$. We further show that, for $(p,2p)$-normic bundles with $p=2$ or $3$, it is enough to assume that the extension degree is divisible by $p$. We also prove that the divisibility hypothesis is, in general optimal, by constructing a conic bundle for which the Brauer--Manin obstruction persists over a quadratic extension.