On the quasi group of a cubic surface over a finite field

We construct nontrivial homomorphisms from the quasi group of some cubic surfaces over \bbF_{\!p} into a group. We show experimentally that the homomorphisms constructed are the only possible ones and that there are no nontrivial homomorphisms in the other cases. Thereby, we follow the classification of cubic surfaces, due to A. Cayley

On the order three Brauer classes for cubic surfaces

We describe a method to compute the Brauer-Manin obstruction for smooth cubic surfaces over \bbQ such that \Br(S)/\Br(\bbQ) is a 3-group. Our approach is to associate a Brauer class with every ordered triplet of Galois invariant pairs of Steiner trihedra. We show that all order three Brauer classes may be obtained in this way. To show the effect of the obstruction, we give explicit examples.Comment: Extended introduction, discussion of the example of Cassels and Gu