Extendability of automorphisms of K3 surfaces

TL;DR

This paper investigates conditions under which automorphisms of K3 surfaces over p-adic fields can be extended to smooth models, focusing on finite order symplectic automorphisms and providing examples of non-extendable actions.

Contribution

It establishes that finite order symplectic automorphisms of K3 surfaces are extendable over good reduction models, using birational geometry and equivariant resolutions.

Findings

Automorphisms of finite order prime to p and symplectic are extendable.

Examples of automorphisms that are not extendable are provided.

The proof employs birational geometry and equivariant resolutions.

Abstract

A K3 surface $X$ over a $p$-adic field $K$ is said to have good reduction if it admits a proper smooth model over the ring of integers of $K$. Assuming this, we say that a subgroup $G$ of $\mathrm{Aut}(X)$ is extendable if $X$ admits a proper smooth model equipped with $G$-action (compatible with the action on $X$). We show that $G$ is extendable if it is of finite order prime to $p$ and acts symplectically (that is, preserves the global $2$-form on $X$). The proof relies on birational geometry of models of K3 surfaces, and equivariant simultaneous resolutions of certain singularities. We also give some examples of non-extendable actions.