A uniform construction of smooth integral models and a recipe for computing local densities

TL;DR

This paper presents a unified method for constructing smooth integral models of certain lattices over local fields and provides a practical recipe for calculating local densities, crucial for classifying algebraic forms.

Contribution

It introduces a simple, uniform construction of smooth integral models for various lattices and proposes a new approach to compute local densities systematically.

Findings

A uniform construction method for smooth integral models

A practical recipe for computing local densities

Introduction of a conjecture on rational points of special fibers

Abstract

In this paper, 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 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.