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 $S(\mathbb{Q})$?



Globalising tori and weak approximation
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](https://academic.oup.com/imrn/article/2001/23/1229/914385). 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.


