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Is it possible to construct the Kuga-Satake abelian variety attached to a K3 surfaces (over a local field) only using p-adic methods?

If the K3 surface is defined over a local field, the Kuga-Satake abelian variety is defined over the same local field? over a finite extension?

Where can I look for work/results in this direction?

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A good place to look is the paper "Kuga-Satake abelian varieties of K3 surfaces in mixed characteristic". J. Reine Angew. Math. 648 (2010), 13–67, by Jordan Rizov. There are also related results by Yves Andre which are mentioned in Rizov's paper.

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