Let $F$ be a number field, $p\in\mathbb{Z}$ a prime which is unramified in $F$ and $G$ a connected reductive group over $F$. Moreover $G$ is supposed to be quasi-split over $p$.

Does there exist a finite set of primes $S\subset\mathbb{Z}$ which does not contain $p$ and a split reductive group $\mathcal{G}$ defined over $\mathbb{Z}[1/S]$ such that for some finite extension $K/F$ which is not ramified over $p$ we have the following isomorphism:

$$ \mathcal{G}\times_{\mathbb{Z}[1/S]}K\cong G\times_FK.$$

More generally, given a reductive group $G/F$, does there exist some sort of split integral model $\mathcal{G}$ defined over rings like $\mathbb{Z}$, $\mathbb{Z}[1/S]$ or $\mathcal{O}_F$?


If I understand the question correctly, the answer is no, but I think it can be changed to yes if the word quasi-split is replaced by unramified.

Let E be a quadratic extension of the rational numbers ramified at p. Let G be the special unitary group associated to the quadratic extension E/Q. This is a quasi-split group over Q. It does not split over any extension of Q unramified at p.

In general, a reductive group over a number field always has a reductive integral model over a ring of S-integers by a spreading out argument, since reductivity is an open condition. One cannot however ensure that this integral model is split. Groups in general are split only etale-locally. So you can make your group split over a ring of S-integers in some field extension, but in order to control the ramification, you need a stronger condition than quasi-split.

  • $\begingroup$ Thanks for your answer. What does it mean by saying that $G$ is unramified over $p$? I am sorry, I am not quite familiar with some terminologies in algebraic group. $\endgroup$
    – tanjia
    Sep 21 '19 at 3:36
  • $\begingroup$ A reductive group G over a local field F with ring of integers O is unramified if it is quasi-split and splits after an unramified extension, or equivalently is the generic fibre of a reductive group scheme over O. $\endgroup$ Sep 21 '19 at 4:05

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