ABSTRACT. In this paper we define a relative rigid fundamental group, which associates to a section p of a smooth and proper morphism f : X → S in characteristic p, with dim S = 1, a Hopf algebra in the ind-category of overconvergent F-isocrystals on S. We prove a base change property, which says that the fibres of this object are the Hopf algebras of the rigid fundamental groups of the fibres of f . We explain how to use this theory to define period maps as Kim does for varieties over number fields, and show in certain cases that the targets of these maps can be interpreted as varieties.
CONTENTS
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 ℓ‐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 Hnormalét2false(XK¯,Qℓfalse) 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.
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