Local-global principle for certain biquadratic normic bundles

Abstract

Let $X$ be a proper smooth variety having an affine open subset defined by the normic equation $N_{k(\sqrt{a},\sqrt{b})/k}(x)=Q(t_1,...,t_m{)}^2$ over a number field $k$. We prove that : (1) the failure of the local-global principle for zero-cycles is controlled by the Brauer group of $X$; (2) the analogue for rational points is also valid assuming Schinzel's hypothesis.

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