Say that $G$ is a reductive group over a local field $F$ and $g\in G(F)$. Can we show that the conjugacy class of $g$ in $G(F)$ has finite index in the $F$-rational part of the conjugacy class of $g$ in $G(\overline{F})$?
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1$\begingroup$ What do you mean by finite index? Do you mean that the $F$-part of the conjugacy class of $g$ in $G(\bar{F})$ splits into finitely many $G(F)$-orbits? $\endgroup$– YCorCommented Dec 10, 2013 at 14:00
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2$\begingroup$ This holds (in much greater generality) by Corollaire 6.4 in Borel-Serre "théorèmes de finitude en cohomologie galoisienne", CMH 1964, when $F$ (locally compact) has characteristic 0. This greater generality fails for local fields of finite characteristic. But I don't know about your specific statement. $\endgroup$– YCorCommented Dec 10, 2013 at 14:04
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1$\begingroup$ I'd add to what Yves says a probably useful reference to an old paper by R. Kottwitz, "Ratiopnal conjugacy classes in reductive groups", Duke Math. J. 49 (1982). Probably there is further work along these lines in later papers, too. Whatever "finite index" means in your question, related matters have certainly been worked out over local fields. It would help to provide more context for your question, given the substantial history. $\endgroup$– Jim HumphreysCommented Dec 10, 2013 at 14:36
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2$\begingroup$ Let $G$ act on itself by conjugation, so you ask if $Q:=(G.g)(F)/G(F).g$ is finite. Since $G\rightarrow G.g$ is a right $Z_G(g)$-torsor (for etale topology if $Z_G(g)$ is smooth, fppf in general), by Cor. 1 to Prop. 36 in section 5.4 of Ch. I of Serre's "Galois cohomology" and its fppf variant we see that $Q=\ker({\rm{H}}^1(F,Z_G(g))\rightarrow {\rm{H}}^1(F,G))$. By 4.3 in Ch. III of that book, ${\rm{H}}^1(F,H)$ is finite if $F$ has characteristic 0; if char$(F)>0$ then finiteness holds if $H$ is connected reductive (rests on Bruhat-Tits) but fails in the smooth connected commutative case. $\endgroup$– user76758Commented Dec 10, 2013 at 15:39
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$\begingroup$ Thanks very much, can you recommend any good reference for reading about torsors? $\endgroup$– RupertCommented Dec 12, 2013 at 12:24
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