Title: Integral models of reductive groups and integral Mumford-Tate groups,
Abstract: 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
$\Lambda$ in the vector space $V$ defines a model $\hat{G}_{\Lambda}$ of $G$
over $\mathscr{O}_K$. One may wonder to what extent $\Lambda$ is determined by
the group scheme $\hat{G}_{\Lambda}$. In this paper we prove that up to a
natural equivalence relation on the set of lattices there are only finitely
many $\Lambda$ corresponding to one model $\hat{G}_{\Lambda}$. Furthermore, we
relate this fact to moduli spaces of abelian varieties as follows: let
$\mathscr{A}_{g,n}$ 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 $\mathscr{A}_{g,n}$ with a given
integral generic Mumford-Tate group.