# Field extensions that preserve given cohomology classes

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.

• 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. – Mikhail Borovoi Jun 2 '20 at 15:13
• 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. – Mikhail Borovoi Jun 2 '20 at 15:31
• 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! – Mikhail Borovoi Jun 2 '20 at 16:03