Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Picard ranks of K3 surfaces over function fields and the Hecke orbit
  conjecture   Let $\mathscr{X} \rightarrow C$ be a non-isotrivial and generically ordinary
family of K3 surfaces over a proper curve $C$ in characteristic $p \geq 5$. We
prove that the geometric Picard rank jumps at infinitely many closed points of
$C$. More generally, suppose that we are given the canonical model of a Shimura
variety $\mathcal{S}$ of orthogonal type, associated to a lattice of signature
$(b,2)$ that is self-dual at $p$. We prove that any generically ordinary proper
curve $C$ in $\mathcal{S}_{\overline{\mathbb{F}}_p}$ intersects special
divisors of $\mathcal{S}_{\overline{\mathbb{F}}_p}$ at infinitely many points.
As an application, we prove the ordinary Hecke orbit conjecture of Chai--Oort
in this setting; that is, we show that ordinary points in
$\mathcal{S}_{\overline{\mathbb{F}}_p}$ have Zariski-dense Hecke orbits. We
also deduce the ordinary Hecke orbit conjecture for certain families of unitary
Shimura varieties.

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The categories of this paper are: acc-phys
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The categories of this paper are: astro-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: supr-con
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The categories of this paper are: nlin
23
The categories of this paper are: solv-int
3
The categories of this paper are: physics
32
The categories of this paper are: atom-ph
12
The categories of this paper are: nucl-th
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The categories of this paper are: q-fin
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The categories of this paper are: dg-ga
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The categories of this paper are: eess
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The categories of this paper are: quant-ph
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The categories of this paper are: funct-an
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The categories of this paper are: cs
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The categories of this paper are: hep-lat