On good reduction of some K3 surfaces related to abelian surfaces

TL;DR

This paper extends the Neron--Ogg--Safarevic criterion to certain K3 surfaces related to abelian surfaces, providing new criteria for good reduction using Galois actions on cohomology, including p-adic analogues.

Contribution

It establishes an analogue of the Neron--Ogg--Safarevic criterion for specific K3 surfaces with Shioda--Inose structures, linking their good reduction to Galois representations.

Findings

Proves criterion for good reduction of K3 surfaces related to abelian surfaces.

Includes p-adic analogue of the reduction criterion.

Incorporates Ito's unpublished results on Kummer surfaces.

Abstract

The Neron--Ogg--Safarevic criterion for abelian varieties tells that whether an abelian variety has good reduction or not can be determined from the Galois action on its l-adic etale cohomology. We prove an analogue of this criterion for some special kind of K3 surfaces (those which admit Shioda--Inose structures of product type), which are deeply related to abelian surfaces. We also prove a p-adic analogue. This paper includes Ito's unpublished result for Kummer surfaces.