Timeline for twisted forms of a given group embedded in a second group?
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12 events
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| Nov 11, 2010 at 8:42 | comment | added | genshin | thanks! How can I tag this comment as an answer? Or edit some of them into one? | |
| Nov 5, 2010 at 18:02 | comment | added | BCnrd | 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 ${\rm{GL}}_n$. | |
| Nov 5, 2010 at 17:33 | comment | added | genshin | 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 $GL_2$. | |
| Nov 5, 2010 at 17:10 | comment | added | BCnrd | 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 $H/Z_H$). | |
| Nov 5, 2010 at 17:06 | comment | added | BCnrd | 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.) | |
| Nov 5, 2010 at 17:05 | history | edited | genshin | CC BY-SA 2.5 |
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| Nov 5, 2010 at 16:54 | comment | added | genshin | 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. | |
| Nov 5, 2010 at 16:42 | history | edited | genshin | CC BY-SA 2.5 |
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| Nov 5, 2010 at 16:39 | history | edited | BCnrd | CC BY-SA 2.5 |
Replaced H' with L in (2) (typo)
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| Nov 5, 2010 at 16:39 | history | edited | genshin | CC BY-SA 2.5 |
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| Nov 5, 2010 at 16:34 | history | edited | BCnrd | CC BY-SA 2.5 |
Replaced G with H at end of 2nd paragraph (typo)
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| Nov 5, 2010 at 16:21 | history | asked | genshin | CC BY-SA 2.5 |