Timeline for $\mathbb{Q}$-forms of $\mathrm{SL}_2\times \mathrm{SL}_2$

8 events

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Dec 5, 2014 at 14:30	comment	added	user74230		YCor's list is exhaustive. In general, the (connected and) simply connected semisimple $k$-groups are precisely the direct products of Weil restrictions $\prod_i {\rm{R}}_{k'_i/k}(G'_i)$ for finite separable extensions $k'_i/k$ and an absolutely simple simply connected semisimple $k'_i$-group $G'_i$ for each $i$. Here, $\sum [k'_i:k]$ is the number of simple factors of $G_{k_s}$, so in your case $\sum [k'_i:k]=2$, recovering the options listed by YCor. The idea of the proof is that ${\rm{Gal}}(k_s/k)$ permutes the simple factors of $G_{k_s}$.
Dec 5, 2014 at 14:07	comment	added	ThiKu		The first case is to some extent discussed in the examples section of de.wikipedia.org/wiki/Arithmetische_Gruppe#Beispiele
Dec 5, 2014 at 12:04	comment	added	YCor		There are at least these ones: The non-$Q$-simple ones are the $G_1\times G_2$ where $G_1,G_2$ are $Q$-forms of $SL_2$. The $Q$-simple ones (or some of them?) are the Weil restrictions "$G(K)$" where $G$ is a form of $SL_2$ and $K$ is a quadratic extension of $Q$.
S Dec 5, 2014 at 11:45	history	edited	Ben McKay	CC BY-SA 3.0	corrected spelling and grammar
S Dec 5, 2014 at 11:45	history	suggested	jmc	CC BY-SA 3.0	corrected spelling and grammar
Dec 5, 2014 at 11:44	review	Suggested edits
S Dec 5, 2014 at 11:45
Dec 5, 2014 at 11:37	review	First posts
Dec 5, 2014 at 11:44
Dec 5, 2014 at 11:30	history	asked	C.C.	CC BY-SA 3.0