Finite group actions, rational fixed points and weak Néron models

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 Néron models.

Proposition 4.2 added and results in Section 4 generalized