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Picard rank jumps for K3 surfaces with bad reduction

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arxiv 2203.09559 v2 pith:YU5IA4TS submitted 2022-03-17 math.NT math.AG

classification math.NTmath.AG
keywords jumpsordinarypicardpreviousproverankreductionsurfaces
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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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  1. Monodromy results for abelian surfaces and K3 surfaces with bad reduction

    math.NT 2024-11 reject novelty 6.0 of 10

    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...

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