A NéronOggShafarevich criterion for K3 surfaces
Abstract
The naive analogue of the NéronOggShafarevich 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 semistable 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 étale cohomology.
 Publication:

arXiv eprints
 Pub Date:
 January 2017
 arXiv:
 arXiv:1701.02945
 Bibcode:
 2017arXiv170102945C
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Number Theory;
 14G20;
 14F20;
 14F30
 EPrint:
 52 pages, completely rewritten with significantly stronger main results