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.
Picard rank jumps for K3 surfaces with bad reduction
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abstract
Let $X$ be a K3 surface over a number field. We prove that $X$ has infinitely many specializations where its Picard rank jumps, hence extending our previous work with Shankar--Shankar--Tang to the case where $X$ might have potentially bad reduction. We prove a similar result for generically ordinary non-isotrivial families of K3 surfaces over curves over $\overline{\mathbb{F}}_p$ which extends previous work of Maulik--Shankar--Tang. As a consequence, we give a new proof of the ordinary Hecke orbit conjecture for orthogonal and unitary Shimura varieties.
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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.