Timeline for Forms of ${\rm SL}(2)$

11 events

when toggle format	what		by	license	comment
Dec 7, 2020 at 9:31	history	edited	David Loeffler	CC BY-SA 4.0	added 85 characters in body
Dec 6, 2020 at 16:50	comment	added	Mikhail Borovoi		@YCor: Yes, $k$-forms of ${\rm SL}_2$ $\leftrightarrow$ (quaternion algebras over $k$) holds for arbitrary fields. See Serre, Galois Cohomology, Section III.1.4.
Dec 6, 2020 at 16:32	comment	added	Mikhail Borovoi		Excellent! Thank you!
Dec 6, 2020 at 16:30	vote	accept	Mikhail Borovoi
Dec 6, 2020 at 15:26	history	edited	LSpice	CC BY-SA 4.0	Various \DeclareMathOperator's
Dec 6, 2020 at 13:23	comment	added	YCor		What are restrictions on $k$ to make the Galois cohomology argument work? It works for $k$ perfect, and probably in more cases here? I guess that the result itself (at least: $k$-forms of $\mathrm{SL}_2\leftrightarrow$ (quaternion algebras over $k$) holds for arbitrary fields?
Dec 6, 2020 at 13:17	history	edited	David Loeffler	CC BY-SA 4.0	edited body
Dec 6, 2020 at 13:17	comment	added	David Loeffler		Aargh, of course you are correct, for $n > 2$ there is a quasi-split outer form. I will edit to make the statement for $n = 2$ only.
Dec 6, 2020 at 13:06	comment	added	Jef		Small unimportant correction: it is not true that the root datum of $SL_n$ has no non-trivial automorphisms if $n>2$: in that case the Dynkin diagram has a unique nontrivial involution as a symmetry.
Dec 6, 2020 at 10:53	history	edited	David Loeffler	CC BY-SA 4.0	added 376 characters in body
Dec 6, 2020 at 10:26	history	answered	David Loeffler	CC BY-SA 4.0