Finite group actions, rational fixed points and weak N\'eron models

TL;DR

This paper extends Serre's result on rational fixed points for finite group actions on affine spaces over finite fields to more general local fields, using the theory of weak Néron models.

Contribution

It generalizes the existence of rational fixed points to henselian discretely valued fields with various residue fields by analyzing group actions on weak Néron models.

Findings

Existence of rational fixed points over certain local fields.

Extension of Serre's result to more general residue fields.

Application of weak Néron models to group action analysis.

Abstract

If $G$ is a finite $\ell$-group acting on an affine space $\mathbb{A}^n$ over a finite field $K$ of cardinality prime to $\ell$, Serre has shown that there exists a rational fixed point. We generalize this to the case where $K$ is a henselian discretely valued field of characteristic zero with algebraically closed residue field and with residue characteristic different from $\ell$. We also treat the case where the residue field is finite of cardinality $q$ such that $\ell$ divides $q-1$. To this aim, we study group actions on weak N\'eron models.