The theorem of Matthews, Vaserstein and Weisfeiler asserts that if $G$ is a simply connected absolutely almost simple groups over $Q$ and $\Gamma$ a finitely generated subgroup of $G(Q)$ which is Zariski dense in $G$ then for almost all primes $p$, the reduction of $\Gamma$ mod $p$ is $G_p(F_p)$.

I was wondering wether the assumption `absolutely almost simple' is really necessary? For example, would the conclusion hold for a group like $Res_{F/Q} G$ where $F$ is a number field and $G$ absolutely simple simply connected over $F$?

  • 1
    $\begingroup$ The second paper, by Weisfeiler alone (Annals of Mathematics, Second Series, Vol. 120, No. 2 (Sep., 1984), pp. 271-315), addresses the semisimple case. The statement is more delicate then, in part because of the subtleties mentioned in the cw answer below. $\endgroup$
    – YCor
    Nov 4 '13 at 20:56
  • $\begingroup$ @Yves: It would be of interest to determine if the (necessary, at least in char. 0) trace condition imposed there is a consequence of a Zariski-density hypothesis on the Weil restriction. Also, that paper does still keep the simply connected hypothesis (as is certainly reasonable). $\endgroup$
    – Marguax
    Nov 6 '13 at 19:24

This isn't an answer, but rather an extended comment to recast the question from another perspective that is given "more directly" over number fields. Perhaps this is something the OP is already well aware of, but hard to tell from the formulation of the question. Hopefully someone else can address the punchline at the end.

In Theorem 8.1 of the paper by those 3 authors, one finds a strong approximation refinement (which immediately implies the "one prime at a time residual surjectivity almost everywhere" property): for a sufficiently large finite set $S$ of places of $\mathbf{Q}$ containing the archimedean place, and $\mathbf{A}^S$ the factor ring of adeles away from $S$, the closure of $\Gamma$ in $G(\mathbf{A}^S)$ is open.

This adelic strengthening seems like the right statement to be trying to generalize over number fields $F$: if $G$ if a connected semisimple $F$-group that is absolutely simple over $F$ and simply connected, and if $\Gamma \subset G(F)$ is a finitely generated subgroup, then under what "algebro-geometric density" conditions on $\Gamma$ can we conclude that the closure of $\Gamma$ in $G(\mathbf{A}_F^S)$ is open for a suitable finite set $S$ of places of $F$ (containing the archimedean places)?

[In general, the connected semisimple $\mathbf{Q}$-groups that are simply connected are precisely the finite products $\prod {\rm{R}}_{F_i/\mathbf{Q}}(G_i)$ for number fields $F_i$ and connected semisimple $F_i$-groups $G_i$ that are absolutely simple and simply connected. So the Weil restrictions in the question are exactly the $\mathbf{Q}$-simple cases; i.e., you're asking to relax "absolutely simple" to "$\mathbf{Q}$-simple".]

So let $H = {\rm{R}}_{F/\mathbf{Q}}(G)$ for such $G$ over a number field $F$, and let $\Gamma$ be a finitely generated subgroup of $H(\mathbf{Q}) = G(F)$. If $\Gamma$ is Zariski-dense in $H$ then it is Zariski-dense in $G$, since $\Gamma$ viewed over $\mathbf{Q}$ visibly factors through the Weil restriction to $\mathbf{Q}$ of its Zariski closure in $G$ over $F$. However, the converse is false, as we see using $\Gamma = {\rm{SL}}_n(\mathbf{Z})$ and $G = {\rm{SL}}_n$ over any number field $F \ne \mathbf{Q}$ (for which $\Gamma$ viewed over $\mathbf{Q}$ has Zariski closure equal to the evident $\mathbf{Q}$-subgroup ${\rm{SL}}_n \hookrightarrow {\rm{R}}_{F/\mathbf{Q}}({\rm{SL}}_n)$).

The preceding counterexample shows that the naive analogue over number fields for the absolutely simple case (using Zariski-density over the number field) is false: the subgroup ${\rm{SL}}_n(\mathbf{Z})$ in $G(F)$ for $G = {\rm{SL}}_n$ and a number field $F \ne \mathbf{Q}$ fails to map onto ${\rm{SL}}_n(O_F/\mathfrak{p})$ for any prime $\mathfrak{p}$ with residual degree $> 1$ over $\mathbf{Q}$.

So in effect, your question is asking whether Zariski-density in the Weil restriction to $\mathbf{Q}$ is the "correct" strengthening of Zariski-density over $F$ to imply openness of the closure in $G(\mathbf{A}_F^S)$ of a finitely generated subgroup $\Gamma \subset G(F)$, for some finite set $S$ of places of $F$ (containing the archimedean places). A quick look on MathSciNet doesn't seem to lead to papers that address this, but perhaps someone more familiar with the relevant literature can point to a suitable reference?


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