Local triviality of Galois cohomology classes over $\mathbb{Q}$

Question

Let $A$ be a $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$-module which is a finitely generated free $\mathbb{Z}$-module. I'm interested in the behaviour of cohomology classes in $$\mathrm{H}^1(\mathbb{Q}, A)$$ upon extension to $\mathbb{Q}_p$. In particular, the set of primes $p$ for which a cohomology class $c$ trivialises upon base-change to $\mathbb{Q}_p$ (I denote the base-change of $c$ to $\mathbb{Q}_p$ by $c \otimes \mathbb{Q}_p \in \mathrm{H}^1(\mathbb{Q}_p, A)$).

Let $c \in \mathrm{H}^1(\mathbb{Q}, A)$. Does the set of primes $p$ for which $$c \otimes \mathbb{Q}_p = 0$$ have a density?

Remarks:

• Each $c$ trivialises after some finite extension $k/\mathbb{Q}$. Thus $c \otimes \mathbb{Q}_p = 0$ for any prime $p$ which is completely split in $k$. This gives a positive density of primes $p$ which trivialise $c$; I want to understand all of them.

I understand the problem well in other, similar looking, situations (i.e. for different choices of $A$). For example:

• The answer to the analogous question is true when $A$ is finite; this follows from the Chebotarev density theorem.
• If $A$ is instead the set of algebraic points on an abelian variety the answer is again yes; in fact one has $c \otimes \mathbb{Q}_p = 0$ for all but finitely many primes $p$ by Lang-Weil and Hensel's lemma, as follows by interpreting $c$ as the class of a torsor for an abelian variety.

Answer 1

I think it follows again from Chebotarev. Represent the cohomogy class as an extension of the trivial representation by $A$. Such an extension is itself a Galois action on a finitely-generated $\mathbb Z$-module and thus has finite image. For unramified primes, whether this extension splits over $\mathbb Q_p$ depends only on the conjugacy class of Frobenius in this finite image group.