Weil pairing, fixed field of a $p$-adic Galois representation

Question

Let $A$ be an abelian variety over a $p$-adic field $K$. If $K(A_{p^\infty})$ is the field extension of $K$ obtained by adjoining the coordinates of all $p$-power division points of $A$. By the Weil pairing, it is known that $K(A_{p^\infty})$ contains the field $K(\mu_{p^\infty})$, the field obtained by adjoining to $K$ all the $p$-power roots of unity.

Now, let $X$ be a proper smooth variety over $K$ and consider the $p$-adic Galois representation $$ \rho : G_K \rightarrow GL(V),$$ where $G_K:=\text{Gal}(\bar{K}/K)$ and $V = H^i_{et} (X_{\bar{K}}, \mathbb{Q}_p)$. Let $F := {\bar{K}}^{\text{ker}\rho}$ be the fixed subfield of $\bar{K}$ by the kernel of $\rho$.

Question 1: Does $F$ contain $K(\mu_{p^\infty})$ in this general case?

By comparing with the abelian variety case, I think that for the answer to Question 1 to be yes, one needs to generalize the Weil pairing.

Question 2: Is there a "generalized Weil pairing" for proper smooth varieties $X$ given above?

Answer 1

The answer to question 1 as states is no. For example if $i=0$, $V$ will be the trivial Galois representation (assuming your variety to be geometrically connected).

But the answer to question 2 is yes (which implies that some corrected version of question 1 holds as well): a "generalized Weil pairing" is given by the theory of Poincaré duality in étale cohomology: there is a Galois-equivariant pairing $H^i_{et}(X_{\bar K},\mathbb Q_p)\times H^{2n-i}_{et}(X_{\overline K},\mathbb Q_p) \rightarrow \mathbb Q_p(1)$, where $n$ is the dimension of your variety. When $X$ is an elliptic curve, $n=1$, and taking $i=1$ gives you exactly the Weil pairing.

There is more to say about your question 2: there is a Lefschetz isomorphism $H^i_{et} \rightarrow H^{2n-i}_{et}$ which is a also Galois equivariant up to a suitable Tate twist. Hence you get a perfect equivariant pairing $H^i_{et} \times H^i_{et} \rightarrow \mathbb Q_p(i)$. Hence back to your question 1, you see that $\ker \rho$ is contained in the kernel of the power $i$ of the cyclotomic character, hence that your field $F$ contains if $i>0$ a subfield $K'$ of $K(\mu_{p^\infty})$ such that $K(\mu_{p^\infty})$ is finite degree over $K'$. Moreover, if $i$ is relatively prime to $2(p-1)$, the answer to your question 1 is yes, as the $i$-th power of the cyclotomic character has then same kernel as the cyclotomic character itself.