Local André-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.