Let $G/\mathbb{Z}_p$ be a connected reductive group, and let $T_1, \cdots, T_n \subset G_{\mathbb{Q}_p}$ be maximal tori. Suppose that we have precisely one maximal torus in each $G(\mathbb{Q}_p)$-conjugacy class of maximal tori; is it then true that $T_1(\mathbb{Z}_p), \cdots, T_n(\mathbb{Z}_p)$ (topologically) generate $G(\mathbb{Z}_p)$? Here by $T(\mathbb{Z}_p)$ I mean the $\mathbb{Z}_p$ points of the connected component of the identity of the Neron model $\mathcal{T}$ of $T$. As LSpice pointed out in the comments, we need the inclusions $\mathcal{T}_i \subset G$ to be defined over $\mathbb{Z}_p$.

If $G=\operatorname{GL_n}$, then this is true. Our conjugacy classes of maximal tori correspond to commutative semi-simple algebras $C/\mathbb{Q}_p$ of degree $n$ and $M_n(\mathbb{Z}_p)$ is (topologically) generated by $\{\mathcal{O}_C\}_{C \in \mathcal{C}}$, where $\mathcal{C}$ is the set of isomorphism classes of such $C$ (and we make some choice of $\mathcal{O}_C \to M_n(\mathbb{Z}_p)$ for each class). An argument using the $p$-adic exponential map now shows that the (topological) closure $H$ of the group generated by our tori contains $1+p \cdot M_n(\mathbb{Z}_p)$, and then all we have to do is show that $H$ surjects onto $\operatorname{GL}_n(\mathbb{F}_p)$.

More generally, I wonder if there is an analogous statement with $\mathcal{G}/\mathbb{Z}_p$ a parahoric group scheme.

  • $\begingroup$ I guess you want the $T_i$'s in $G$, not just in $G_{\mathbb Q_p}$, to make sense of $T_i(\mathbb Z_p)$. Is it always possible to choose a cross-section of the set of $G(\mathbb Q_p)$-conjugacy classes of tori in $G_{\mathbb Q_p}$, each of which is defined over $\mathbb Z_p$? I'd think that would involve some kind of strong splitness/tameness (on all tori) condition—maybe something like (as investigated by Fintzen) $p \nmid \lvert W\rvert$, or something? $\endgroup$ – LSpice Aug 3 '20 at 21:03
  • $\begingroup$ Also, when speaking of $G = \operatorname{GL}_n$, you seem to switch to speaking of $\operatorname{Lie}(G)$ instead, since you say "$M_n(\mathbb Z_p)$ is (topologically) generated by $\{\mathcal O_C\}_{C \in \mathcal C}$". Is that intentional, or did I misunderstand? $\endgroup$ – LSpice Aug 3 '20 at 21:05
  • $\begingroup$ That is not intentional, but the statement on Lie algebras can be used to prove the group theoretic statement using the exponential map (and some further arguments). $\endgroup$ – Pol van Hoften Aug 4 '20 at 8:00
  • $\begingroup$ I guess I need my tori to be defined over $\mathbb{Z}_p$, I will edit the question. Also, could you explain why we would need all tori to split over tamely ramified extensions? Does it have something to do with properties of Neron models? $\endgroup$ – Pol van Hoften Aug 4 '20 at 8:03
  • $\begingroup$ I don't know that we would need all tori to split over tamely ramified extensions; I just guessed, and may be wildly off base. My point was that I didn't think a random $\mathbb Q_p$-torus could be given a $\mathbb Z_p$-model, but you're probably right—the connected Néron model should do it, I guess. (I'm not really comfortable, despite @BCnrd's heroic efforts ("Reductive group schemes"), with what 'connected reductive group', or even 'torus', means over $\mathbb Z_p$.) $\endgroup$ – LSpice Aug 4 '20 at 10:05

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