Volume 7 (2024)
Lopuhaä-Zwakenberg, Milan
Integral models of reductive groups and integral Mumford–Tate groups
Annales Henri Lebesgue, Volume 7 (2024), pp. 749-786.

Metadata

DOI10.5802/ahl.210
Keywords Reductive groups, integral models, Mumford–Tate groups

Abstract

Let G be a reductive group over a number field or p-adic field, and let V be a faithful representation of G. A lattice Λ in V induces an integral model mdl G (Λ) of G. The first main result of this paper states that up to the action of the normalizer of G, there are only finitely many Λ yielding the same mdl G (Λ). We first prove this for split G via the theory of Lie algebra representations, then for nonsplit G via Bruhat–Tits theory. The second main result shows that in a moduli space of principally polarized abelian varieties, a special subvariety is determined, up to finite ambiguity, by its integral Mumford–Tate group. We obtain this result by applying the first main result to the symplectic representations underlying special subvarieties.


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