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Let $G$ be a semi-simple and simply connected reductive group over $\mathbb{Q}$ and let $T \subset G_{\mathbb{Q}_p}$ be a maximal torus. A classical result of Harder tells us that we can find a maximal torus $S \subset G$ such that $S_{\mathbb{Q}_p}$ is $G(\mathbb{Q}_p)$ conjugate to $T$. In fact we can find infinitely many of them; we are allowed to specificy the $G(\mathbb{Q}_v)$-conjugacy class of $S_{\mathbb{Q}_v}$ for finitely many places $v$ of $\mathbb{Q}$.

Question 1: Can we choose $S$ such that $S$ satisfies weak approximation at $p$, i.e., such that $S(\mathbb{Q})$ is dense in $S(\mathbb{Q}_p)$? If $T$ splits over an unramified extension of $\mathbb{Q}_p$, then any $S$ works (c.f. Proposition 7.8 of Platonov-Rapinchuk). One could hope that by changing $S$ at places away from $p$, one might be able to get rid of the obstruction to weak approximation. Perhaps this is too much to ask for, and in fact the following weaker statement would suffice for the applications that I have in mind.

Question 2: Let $x \in T(\mathbb{Q}_p)$, can we always choose $S$ such that $x$ is contained in the closure of $S(\mathbb{Q})$?

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I think that Question 1 can be answered in the affirmative using (the proof of) Theorem 1 in the paper by Gopal Prasad and Andrei Rapinchuk "Irreducible Tori in Semisimple Groups", IMRN, 2001, No. 23, 1129-1242. The idea is to specialise a generic torus in G and use a result due to Alexander Klyachko (Borovoi provided an independent proof) which implies weak approximation for such tori.

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  • $\begingroup$ Do you mean the paper of Klyachko referred to by Prasad and Rapinchuk? And what text of Borovoi do you mean? $\endgroup$ Sep 12, 2020 at 19:07
  • $\begingroup$ Yes, I meant the paper of Klyachko mentioned by Prasad and Rapinchuk. They also mention your proof, which only exists as a handwritten letter you sent me many years ago. $\endgroup$ Sep 12, 2020 at 19:20
  • $\begingroup$ The question follows from Theorem 1.1.(i) of your link, which says that we make sure that $S$ is "without affect" by changing $S$ at finitely many places away from $p$ (and tori "without affect" have weak approximation). Of course the cited theorem is about absolutely almost simple groups over number fields, but our $G$ will be a product of restrictions of scalars of those. $\endgroup$ Sep 13, 2020 at 15:05

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