# A N\'eron-Ogg-Shafarevich criterion for K3 surfaces

## TL;DR

This paper establishes a criterion for good reduction of K3 surfaces over discretely valued fields, linking unramified cohomology and Galois representations to the existence of good reduction, correcting naive analogues.

## Contribution

It proves a Néron-Ogg-Shafarevich type criterion for K3 surfaces, relating unramified cohomology and Galois representations to good reduction under potential semi-stable reduction assumptions.

## Key findings

- Good reduction characterized by unramified cohomology and Galois representation matching canonical reduction.
- Corrects naive analogue of Néron-Ogg-Shafarevich criterion for K3 surfaces.
- Results extend to p-adic étale cohomology.

## Abstract

The naive analogue of the N\'eron-Ogg-Shafarevich criterion is false for K3 surfaces, that is, there exist K3 surfaces over Henselian, discretely valued fields $K$, with unramified $\ell$-adic \'etale cohomology groups, but which do not admit good reduction over $K$. Assuming potential semi-stable reduction, we show how to correct this by proving that a K3 surface has good reduction if and only if $H^2_{\mathrm{\acute{e}t}}(X_{\overline{K}},\mathbb{Q}_\ell)$ is unramified, and the associated Galois representation over the residue field coincides with the second cohomology of a certain "canonical reduction" of $X$. We also prove the corresponding results for $p$-adic \'etale cohomology.