The classification of generalised Kummer surfaces in positive characteristic

Building on earlier work of Katsura and Rybakov, we complete the classification, in every characteristic, of all groups $G$ acting on an abelian surface $A$ by automorphisms preserving the group law such that the resolution of the quotient $A\mathrm{/}G$ is a K3 surface. In order to do so, we study actions of groups with $p\mid \mathrm{\mid }G\mathrm{\mid }$ in characteristics $p=2,3$ and $5$. Using the theory of rational double points in positive characteristic, we show that the possible ADE singularity types of $A\mathrm{/}G$ are constrained by the requirement that their local fundamental group contains $G$ as a subgroup, and we determine the singular locus of $A\mathrm{/}G$ via the action of $G$ on the $\mathrm{\ell }$-adic Tate module of $A$. As a key step in the classification, we prove that if $A$ is a supersingular abelian surface and $p\mid \mathrm{\mid }G\mathrm{\mid }$, then $A\mathrm{/}G$ can never be a generalised Kummer surface. Finally, we construct explicit examples of such generalised Kummer surfaces as quotients of products of two elliptic curves.