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Consider the following question about forms of a given group that are embedded in a fixed group.

Fix for simplicity $k$ a perfect field, and $H\subsetneq G$ a pair of connected reductive $k$-groups, with $G$ being $k$-simple. It is possible that one finds other $k$-groups $L\subset G$ that are $k$-forms of $G$.

(1) How to classify those $k$-forms $L$ embedded in $G$ such that $L_\bar{k}$ is conjugate to $H_{\bar{k}}$ by $G(\bar{k})$?

(2) Given a $k$-form $L$ of $H$, can one always find a larger $k$-group $G$ containing both $H$ and $H'$, such that (1) holds for the triple $(H,L,G)$? Here $G$ is required to not normalize any $k$-factor of $H$.

To algebraize the formulation, take $L$ be a $k$-form of $H$ embedded also in $G$, and set the following $k$-groups: $N=N(H,G)$ the normalizer, and $T=T_G(H,L)$ the transporter from $H$ to $L$ in $G$. According to Demazure-Gabriel, Groupes algebriques, II, sect.1, 3.6, 3.7, $T$ is representable in this case and $T$ has a right action by $N$. Evaluating at $\bar{k}$ one sees that $T$ is a $N$-torsor for the etale topology. $T$ admits a $K$-rational point, $K$ being some finite extension of $k$, if and only if the $N$-torsor splits over $K$.

But the above arguments rely on the existence of a $k$-form $L$ embedded in $G$. A priori I don't know if such $k$-forms exists, because given a $\bar{k}$-subgroup of $G_\bar{k}$ conjugate to $H_\bar{k}$, I don't know if there is a criterion to assure that it is also defined over $k$. Such criteria provided, the question can be reduced to

(i) finding $k$-splitting of the torsor $T$;

(ii) classifying the torsors $T$ that come from $k$-forms embedded in $G$.

although (ii) is more or less repeating (1).

Many thanks indeed.

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  • $\begingroup$ I would like to also include the case where $H$ is assumed to be reductive, but the case $H$ being semi-simple is already interesting to me. When one considers $k$-tori embedded in $GL_2$, one finds them of them of the form $Res_{K/k}\mathbb{G}_\mathrm{m}$, with $K$ some etale $k$-algebra of degree 2 embedded in $M_2$. It is true that in general no such description is available, but I wonder in concrete examples, like $GSp_N$, $GU(p,q)$ there are positive examples. $\endgroup$
    – genshin
    Commented Nov 5, 2010 at 16:54
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    $\begingroup$ For any $k$, use $k_s$ in (1). Let $N$ be normalizer of $H$ in $G$ (i.e., $N(R)$ is $G(R)$-normalizer of $H_R$ in $G_R$ for $k$-alg. $R$; exists as closed $k$-subgp scheme in $G$ since $H$ smooth). For $g\in G(k_s)$, $gH_{k_s}g^{-1}$ is Gal-stable iff $\sigma(g)\in gN(k_s)$ for all $\sigma$ in Gal($k_s/k$); i.e.,$g^{-1}\sigma(g)\in N(k_s)$. But $g$ only matters up to rt mult. by $N(k_s)$, so (1) seems to be kernel of map of pointed sets ${\rm{H}}^1(k_s/k,N) \rightarrow {\rm{H}}^1(k_s/k,G)$. (For max. smooth closed $k$-subgp $N'$ in $N$ have $N'(k_s)=N(k_s)$; $N'=N_{\rm{red}}$ if $k$ perfect.) $\endgroup$
    – BCnrd
    Commented Nov 5, 2010 at 17:06
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    $\begingroup$ In (1) it seems that you're asking to classify $L$ including their embeddings into $G$, but in your comment it sounds like you may be interested in just the abstract $k$-isomorphism classes of such $L$. If the latter, then of course take the kernel in my previous comment and pass to its image under the map ${\rm{H}}^1(k_s/k,N) \rightarrow {\rm{H}}^1(k_s/k,\underline{\rm{Aut}}_{H/k})$ into the cohomology of the automorphism scheme of $H$ (whose identity component is $H/Z_H$). $\endgroup$
    – BCnrd
    Commented Nov 5, 2010 at 17:10
  • $\begingroup$ And I wonder if this can be done more effectively. Take $k=\mathbb{Q}$. Say in the case of maximal non-split $k$-tori embedded in $GL_2$, one finds them associated to quadratic extensions $K/k$; if one puts the (reduced) discriminant $d_K$ of the extension as an invariant, then within a given interval $J\subset\mathbb{R}$, there are only finitely many $k$-isomorphic classes of such tori with $d_K\in J$. When $k$ is a general number field, I don't know how to topologize a non-abelian $H^1(k,G)$, but I wonder if there are ways to bound them in an effective way as one does for tori in $GL_2$. $\endgroup$
    – genshin
    Commented Nov 5, 2010 at 17:33
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    $\begingroup$ Dear algchen: I don't think $d_K$ or topologizing cohomology are good ideas. Using simply connected cover $G'$ of the derived group and maximal central torus $Z$ (for which there's a central isogeny $G' \times Z \rightarrow G$), relate things to cohomology of tori and finite $k$-groups of mult. type (& apply CFT). The main ingredients are Hasse principle in the simply connected case and vanishing of degree-1 cohomology in the simply connected case over non-archimedean local fields. This approach immediately brings out degree-$n$ etale algebras ($H^1(k,S_n)$!) for max. tori in ${\rm{GL}}_n$. $\endgroup$
    – BCnrd
    Commented Nov 5, 2010 at 18:02

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