A Néron-Ogg-Shafarevich criterion for K3 surfaces

Abstract

The naive analogue of the Néron-Ogg-Shafarevich criterion is false for K3 surfaces, that is, there exist K3 surfaces over Henselian, discretely valued fields $K$, with unramified $\ell$-adic étale 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 $H_{\acute{\mathrm{e}}\mathrm{t}}^2(X_{\overline{K}},{\mathbb{Q}}_{\ell })$ 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 étale cohomology.

52 pages, completely rewritten with significantly stronger main results