I gave this talk in the conference Young Researchers in Algebraic Number Theory (Y-RANT) on the 4th of September 2025, at the University of Nottingham. This talk is based on the last section of my paper The classification of generalised Kummer surfaces in positive characteristic (Gonzalez-Hernandez, 2026).
References
2026
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The classification of generalised Kummer surfaces in positive characteristic
Alvaro Gonzalez-Hernandez
arXiv preprint, 2026
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/G is a K3 surface. In order to do so, we study actions of groups with p∣|G| 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/G are constrained by the requirement that their local fundamental group contains G as a subgroup, and we determine the singular locus of A/G via the action of G on the \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∣|G|, then A/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.
