Christopher Lazda

The classical Neron-Ogg Shafarevich criterion gives a criterion for good reduction of abelian varieties over a p-adic field K in terms of the Galois action on their Tate module. A similar criterion exists for curves by working instead with the unipotent fundamental group. For K3 surfaces this criterion fails in a rather surprising way: Liedtke and Matsumoto constructed examples of K3 surfaces over Q_p (p >=5) which only admit good reduction over the unique unramified quadratic extension of Q_p. Assuming potential semi-stable reduction, however, they showed that this is the only way it can fail. Namely, if X/K is a K3 surface and the inertia group I_K acts trivially on the etale cohomology of X, then X admits good reduction over a finite and unramified extension of K. Still assuming potential semi-stable reduction, we explain how to improve on this by showing that a K3 surface X/K has good reduction if and only if its second etale cohomology group is unramified, and moreover coincides (as a Galois representation) with the etale cohomology of a certain “canonical reduction” of X. This is joint work with B. Chiarellotto and C. Liedtke.