Finite Groups of Symplectic Automorphisms of Supersingular K3 surfaces in Odd Characteristics

TL;DR

This paper investigates symplectic automorphisms of supersingular K3 surfaces in odd characteristics, introducing new lattice-theoretic tools to classify finite group actions beyond tame cases.

Contribution

It develops the concept of p-root pairs to analyze symplectic group actions on supersingular K3 surfaces, extending previous results to non-tame automorphisms.

Findings

Provides an upper bound for the exponent of p in |G|

Introduces p-root pairs related to root systems and Weyl groups

Offers alternative proofs for existing classification results

Abstract

In 2009, Dolgachev-Keum showed that finite groups of tame symplectic automorphisms of K3 surfaces in positive characteristics are subgroups of the Mathieu group of degree 23. In this paper, we utilize lattice-theoretic methods to investigate symplectic actions of finite groups G on K3 surfaces in odd characteristics. For supersingular K3 surfaces with Artin invariants at least two, we develop a new machinery called p-root pairs to constrain possible symplectic finite group actions (without the assumption of tameness). The concept of p-root pair is closely related to root systems and Weyl groups. In particular, we provide alternative proof for many results by Dolgachev-Keum and give an upper bound for the exponent of p in |G|.