A Uniform Construction of Smooth Integral Models and a Conjectural Recipe for Computing Local Densities

Abstract

In this article, we explain a simple and uniform construction of a smooth integral model associated to a quadratic, (anti)-hermitian, and (anti)-quaternionic hermitian lattice defined over an arbitrary local field. As one major application, we explain a conjectural recipe for computing local densities case by case, which is an essential factor in the classification of forms as above over the ring of integers of a number field, by introducing one conjecture about the number of rational points of the special fiber of a smooth integral model.