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 Jan 13 awarded Popular Question May 8 awarded Nice Question Feb 27 comment orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$ Dear Dave, sorry, I made a silly mistake. Is there a reference for what you said above, namely that "$G({\Bbb Q}_p)\times{\Bbb Q}^*_p$ has only finitely many orbits on the set of vector-space lattices whenever $G$ is split, semi-simple, and simply connected, and ρ is faithful and irreducible."? This is the case I need. Feb 26 comment orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$ Dear Dave, oh. It seems the following simple argument works. $L=A({\Bbb Q}{\Bbb Q}\times ... {\Bbb Q})$ is vector-space lattice iff $A^{-1} G({\Bbb Q}_p)A=G({\Bbb Q}_p)$, i.e. $A\in GL({\Bbb Q}^n)$ is in the normaliser of $G({\Bbb Q}_p)$. $<=$ is easy; $=>$: $A^{-1}Stab(L)A=G({\Bbb Q}_p)\cap GL({\Bbb Q}^n)$; pass to the closures and use $Stab(L)$ is dense in $G(({\Bbb Q}_p)$: $A^{-1}G(({\Bbb Q}_p)A=cl(A^{-1}Stab(L)A)=cl(G({\Bbb Q}_p)\cap GL({\Bbb Q}^n))\subseteq G({\Bbb Q}_p)$. Feb 26 comment orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$ Dear Dave, thank you! It is interesting and shall take some time for me to digest; but it does solve (probably) my question so I "accepted" your answer. Feb 26 accepted orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$ Feb 25 comment orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$ Thanks! And what about $G=GL_n$, should it not be an exception as well? What I ask is true for $G=GL_n$... Feb 24 revised orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$ added 245 characters in body Feb 24 comment orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$ In this case one needs to consider the action of the group of diagonal matrices with non-zero determinant... Feb 24 comment orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$ Thanks, I've made a few correction. Yes, I consider the action of the group of p-adic points on the subgroups L with these properties. Concerning being dense in either Zariski or p-adic topology, it is unclear what the right question is. Feb 24 revised orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$ added 54 characters in body Feb 24 comment orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$ @Venkataramana, thank you, I corrected that. It does seem that the answer might be implied by the theory of arithmetic subgroups, but it feels the question is standard enough to be somewhere in the literature... Feb 24 awarded Promoter Feb 24 revised orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$ edited body Feb 18 revised orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$ added 194 characters in body Feb 18 revised orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$ edited tags Feb 17 asked orbits of linear algebraic group $G({\Bbb Q}_p)$ acting on subgroups of ${\Bbb Q}_p^n$ Jan 4 awarded Necromancer Oct 17 awarded Necromancer Jul 23 comment Is the theory of categories decidable? On the other hand, there is no reason for the equational fragment of the theory of categories without automorphisms be undecidable, and it is actually decidable, see "mishap.sdf.org/…