For a K3 surface in characteristic \(p\ge 3\), there is a projective model
\(X_R\to {Spec \, R}\) in characteristic \(0\) with Picard number \(1\) over a
geometric generic point. In particular, this model essentially kills all
automorphisms.
There is a supersingular K3
surface in characteristic \(3\), with an automorphism of positive entropy, the
logarithm of a Salem number of degree \(22\), which does not lift to
characteristic \(0\) at all. We construct elliptic K3 surfaces in
characteristic \(p\ge 3\) such that the automorphism group of any lifting to
characteristic \(0\) does not hit the whole automorphism subgroup of the
Mordell-Weil group of the elliptic fibrations and some automorphisms of
positive entropy. (joint work with K. Oguiso)
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