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I am working on questions in linear algebraic groups $k$, where $k$ is a local field of positive characteristic $p$. I would like to exclude some bad behaviour using the assumption that $p$ is sufficiently big. Here is a first precise question:

Q1: Suppose that $G$ is $k$-group which is pseudo-reductive but not reductive. Is there a lower bound on the dimension of $G$ depending on $p$? (Or, alternatively: Choose $n$ such that $G$ embeds into $\operatorname{GL}_n$. Is there a lower bound on $n$ depending on $p$?)

(A bit of background: A linear algebraic group defined over $k$ is called pseudo-reductive, if it has no non-trivial normal unipotent subgroup defined over $k$. If $k$ is not perfect, then pseudo-reductive does not imply reductive: the group might have a unipotent radical which is only defined over an inseparable extension of $k$.)

The book of Conrad-Gabber-Prasad ("pseudo-reductive groups") gives a partial answer to Q1. Every such $G$ is obtained by combining two different constructions. One of them uses restrictions of scalars from inseparable field extensions of $k$ and indeed only yields $G$ of big dimensions. However, the other one consists in starting with a (pseudo-)reductive group $G'$ and "replacing" a Cartan subgroup by an abelian pseudo-reductive group $C$. (In particular, if the resulting group $G$ is abelian, then this construction does not give any information, since one can use $C = G$; indeed, it seems that not much is known about abelian pseudo-reductive groups.) The result of CGP allows to reduce Q1 to:

Q2: Like Q1, but with the additional condition that $G$ is abelian.

In fact, the problem I am working on can be solved in a different way if $G$ is abelian, so the only reason I have to consider abelian pseudo-reductive groups is that they appear in the construction of non-abelian ones. Therefore, even the following would be good enough for me:

Q3: Like Q2, but in addition, $G$ should act on a non-commutative reductive group as required in the construction of CGP.

(I don't know whether this is really an additional restriction.)

Finally, if the answer to these questions is no, then the following would be useful:

  • a classification or any information about abelian pseudo-reductive non-reductive groups $G$ of small dimension (e.g. smaller than $p$)

  • the same, but under the additional condition that $G$ is compact (for this to make sense, recall that I am assuming that $k$ is a local field)

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By Lemma 9.3 of Totaro's paper "Pseudo-abelian varieties" (see the arxiv version), for any field $k=k_s$ and 1-dimensional $k$-wound smooth connected unipotent $k$-group $U$ there is a commutative pseudo-reductive extension of $U$ by $\mathbf{G}_m$. So the minimal possible dimension of 2 is realized over any separably closed field $k$ that isn't algebraically closed (as the "Rosenlicht construction" yields such $U$ over any such $k$). Given $k \ne k_s$, Totaro's construction over $k_s$ descends to some finite separable extension of $k$. Making "$k$-anisotropic" examples seems harder. – user29720 Feb 27 '13 at 23:42
Even though it appears to be hopeless to say anything in general about commutative pseudo-reductive groups, Totaro's paper contains quite a bit of interesting information about commutative pseudo-reductive groups (in that it makes several different kinds of constructions). So it definitely worth a close look (even if it may have the main effect of just convincing you that the commutative case is even more unwieldy than you had hoped). – user29720 Feb 28 '13 at 1:13

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