Local-global principle for certain biquadratic normic bundles

TL;DR

This paper investigates the local-global principle for zero-cycles and rational points on certain biquadratic normic bundles, showing the Brauer group controls failures and assuming Schinzel's hypothesis for rational points.

Contribution

It establishes the Brauer-Manin obstruction as the only obstruction for zero-cycles and extends the result to rational points under Schinzel's hypothesis.

Findings

Failure of the local-global principle is controlled by the Brauer group.

The analogue for rational points holds assuming Schinzel's hypothesis.

Provides new insights into biquadratic normic bundles over number fields.

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.