Integral models of reductive groups and integral Mumford-Tate groups

Let $G$ be a reductive algebraic group over a $p$-adic field or number field $K$, and let $V$ be a $K$-linear faithful representation of $G$. A lattice $\mathrm{\Lambda }$ in the vector space $V$ defines a model ${\widehat{G}}_{\mathrm{\Lambda }}$ of $G$ over ${\mathcal{O}}_K$. One may wonder to what extent $\mathrm{\Lambda }$ is determined by the group scheme ${\widehat{G}}_{\mathrm{\Lambda }}$. In this paper we prove that up to a natural equivalence relation on the set of lattices there are only finitely many $\mathrm{\Lambda }$ corresponding to one model ${\widehat{G}}_{\mathrm{\Lambda }}$. Furthermore, we relate this fact to moduli spaces of abelian varieties as follows: let ${\mathcal{A}}_{g,n}$ be the moduli space of principally polarised abelian varieties of dimension $g$ with level $n$ structure. We prove that there are at most finitely many special subvarieties of ${\mathcal{A}}_{g,n}$ with a given integral generic Mumford-Tate group.