Integral models of reductive groups and integral Mumford–Tate groups

Lopuhaä-Zwakenberg, Milan

Annales Henri Lebesgue, Tome 7 (2024), pp. 749-786

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 $\Lambda$ in $V$ induces an integral model ${\mathtt{mdl}}_G\left(\Lambda \right)$ 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 $\Lambda$ yielding the same ${\mathtt{mdl}}_G\left(\Lambda \right)$. 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.

Soit $G$ un groupe réductif sur un corps de nombres ou un corps $p$-adique, et soit $V$ une représentation fidèle de $G$. Un réseau $\Lambda$ dans $V$ induit un modèle intégral ${\mathtt{mdl}}_G\left(\Lambda \right)$ de $G$. Le premier résultat principal de cet article montre que, à l’action du normalisateur de $G$ près, il n’existe qu’un nombre fini de $\Lambda$ produisant le même ${\mathtt{mdl}}_G\left(\Lambda \right)$. Nous le prouvons d’abord pour $G$ scindé via la théorie des représentations des algèbres de Lie, puis pour $G$ non scindé via la théorie de Bruhat–Tits. Le second résultat principal montre que dans un espace de modules de variétés abéliennes principalement polarisées, une sous-variété spéciale est déterminée, à une ambiguïté finie près, par son groupe de Mumford–Tate intégral. Nous obtenons ce résultat en appliquant le premier résultat principal aux représentations symplectiques sous-jacentes aux sous-variétés spéciales.

Classification : 14D07, 14D22, 20G25, 20G30
Keywords: Reductive groups, integral models, Mumford–Tate groups

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