Odd torsion Brauer elements and arithmetic of diagonal quartic surfaces over number fields

Abstract

We use recent advances in the local evaluation of Brauer elements to study the role played by {\it odd} torsion elements of the Brauer group in the arithmetic of diagonal quartic surfaces over {\it arbitrary} number fields. We show that over a local field if the order of the Brauer element is odd and coprime to the residue characteristic then the evaluation map it induces on the local points is constant. Over number fields we give a sufficient condition on the coefficients of the equation, which is mild and easy to check, so that the odd torsion does not obstruct weak approximation. We also note a systematic way to produce $K3$ surfaces over $\Q_2$ with good reduction and a non-trivial 2-torsion element of the Brauer group with Swan conductor zero.