On Rational Points of Varieties over Local Fields having a Model with Tame Quotient Singularities

TL;DR

This paper investigates rational points on smooth varieties over local fields with models exhibiting tame quotient singularities, providing constructions of weak Néron models, formulas for invariants, and conditions for rational points to exist.

Contribution

It introduces a method to construct canonical weak Néron models for varieties with tame quotient singularities and derives formulas for motivic invariants and rational points.

Findings

Constructed canonical weak Néron models for varieties with tame quotient singularities.

Derived formulas for motivic Serre invariant and rational volume.

Established conditions for the existence of rational points with potential good reduction.

Abstract

We study rational points on a smooth variety X over a complete local field K with algebraically closed residue field, and models of X with tame quotient singularities. If a model of X is the quotient of a Galois action on a weak N\'eron model of the base change of X to a tame Galois extension of K, then we construct a canonical weak N\'eron model of X with a map to this model, and examine its special fiber. As an application we get examples of singular models of X such that X has K-rational points specializing to a singular point of this model. Moreover we obtain formulas for the motivic Serre invariant and the rational volume, and the existence of K-rational points on certain K-varieties with potential good reduction.