Evaluating Azumaya algebras on cubic surfaces

TL;DR

This paper investigates the behavior of Azumaya algebras on cubic surfaces over local and number fields, analyzing local evaluation maps and their dependence on geometric reduction, and demonstrating cases with no Brauer-Manin obstruction.

Contribution

It provides a detailed description of local evaluation maps for Azumaya algebras on cubic surfaces and extends known results by identifying conditions where the Brauer-Manin obstruction vanishes.

Findings

Evaluation maps depend on the geometry of the reduction of the surface.

Certain cubic surfaces with cone reductions have no Brauer-Manin obstruction.

Extension of previous results by Colliot-Thélène, Kanevsky, and Sansuc.

Abstract

Let X be a cubic surface over a local number field k. Given an Azumaya algebra on X, we describe the local evaluation map X(k) -> Q/Z in two cases, showing a sharp dependence on the geometry of the reduction of X. We show that a suitably generic cubic surface over a number field, whose reduction at some prime is a cone, has no Brauer-Manin obstruction. This extends results of Colliot-Th\'el\`ene, Kanevsky and Sansuc.