For ordinary abelian surfaces and K3 surfaces over F_q((t)) with semi-stable bad reduction, the p-adic monodromy image of inertia is described by the reduction type of the abelian quotient: unipotent, finite index, or trivial.
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Monodromy results for abelian surfaces and K3 surfaces with bad reduction
For ordinary abelian surfaces and K3 surfaces over F_q((t)) with semi-stable bad reduction, the p-adic monodromy image of inertia is described by the reduction type of the abelian quotient: unipotent, finite index, or trivial.