Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Local Andr\'{e}-Oort conjecture for the universal abelian variety   We prove a $p$-adic analogue of the Andr\'{e}-Oort conjecture for
subvarieties of the universal abelian varieties containing a dense set of
special points. Let $g$ and $n$ be integers with $n \geq 3$ and $p$ a prime
number not dividing $n$. Let $R$ be a finite extension of $W[{\mathbb
F}_p^{\mathrm alg}]$, the ring of Witt vectors of the algebraic closure of the
field of $p$ elements. The moduli space $\cA = \cA_{g,1,n}$ of $g$-dimensional
principally polarized abelian varieties with full level $n$-structure as well
as the universal abelian variety $\pi:\cX \to \cA$ over $\cA$ may be defined
over $R$. We call a point $\xi \in \cX(R)$ \emph{$R$-special} if
$\cX_{\pi(\xi)}$ is a canonical lift and $\xi$ is a torsion point of its fibre.
We show that an irreducible subvariety of $\cX_R$ containing a dense set of
$p$-special points must be a special subvariety in the sense of mixed Shimura
varieties. Our proof employs the model theory of difference fields.

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The categories of this paper are: funct-an
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The categories of this paper are: physics
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The categories of this paper are: q-bio
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: adap-org
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The categories of this paper are: cond-mat
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The categories of this paper are: patt-sol
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The categories of this paper are: gr-qc
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The categories of this paper are: cmp-lg
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The categories of this paper are: comp-gas
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The categories of this paper are: stat
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The categories of this paper are: chem-ph
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The categories of this paper are: math-ph