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

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.