We determine the odd order torsion subgroup of the Brauer group of diagonal quartic surfaces over the field of rational numbers. We show that a non-constant Brauer element of odd order always obstructs weak approximation but never the Hasse principle.

We exhibit central simple algebras over the function field of a diagonal quartic surface over the complex numbers that represent the 2-torsion part of its Brauer group. We investigate whether the 2-primary part of the Brauer group of a diagonal quartic surface over a number field is algebraic and give sufficient conditions for this to be the case. In the last section we give an obstruction to weak approximation due to a transendental class on a specific diagonal quartic surface, an obstruction which cannot be explained by the algebraic Brauer group which in this case is just the constant algebras.

We obtain an easy sufficient condition for the Brauer group of a diagonal quartic surface D over ℚ to be algebraic. We also give an upper bound for the order of the quotient of the Brauer group of D by the image of the Brauer group of ℚ. The proof is based on the isomorphism of the Fermat quartic surface with a Kummer surface due to Mizukami.

We study local-global principles for zero-cycles on K3 surfaces defined over number fields. We follow an idea of Liang to use the trivial fibration over the projective line.

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