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.