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Let $G/\mathbb{Q}$ be a connected reductive group, let $C \subset H^1(\mathbb{Q}, G)$ be a finite set of points and let $n \ge 2$ be an integer. Is it always possible to find a finite extension $F/\mathbb{Q}$ of degree $n$ such that the map $C \to H^1(\mathbb{Q}, G) \to H^1(F,G_F)$ is injective? This should work for groups satisfying the Hasse principle, using extensions $F$ that totally split at all relevant finite primes and are unramified at infinity.

The case that I am really interested in however, is when $C=\operatorname{Ker}^1(\mathbb{Q},G)$. Moreover, I wonder if it is possible to arrange for $F$ to be totally real and such that a fixed prime $p$ remains inert in $F$. Is this too much to ask for in general? If it helps, all the groups that I am interested in admit Shimura data.

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    $\begingroup$ You can reduce your question to a question about complexes of tori using Theorem 5.11 of Memoirs of the AMS 132 (1998), No. 626. $\endgroup$ – Mikhail Borovoi Jun 2 '20 at 15:13
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    $\begingroup$ To deal with complexes of tori, you can use the paper by Cyril Demarche, Suites de Poitou-Tate pour les complexes de tores à deux termes. Int. Math. Res. Not. IMRN 2011, no. 1, 135–174. See also arxiv.org/abs/0906.3453. $\endgroup$ – Mikhail Borovoi Jun 2 '20 at 15:31
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    $\begingroup$ You can try to try to answer your question for semisimple groups using calculations of $H^2$ with coefiicients in certain finite abelian algebraic groups in Sections 1 and 2 of Sansuc's paper. Good luck! $\endgroup$ – Mikhail Borovoi Jun 2 '20 at 16:03

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