Let $k$ a field with $char(k)=p>0$, separable closure $k^{sep}$ and $f:X\rightarrow Y$ a smooth projective morphism of smooth variety over $k$.

1)Is it true that there exists an open susbset $U$ of $Y$ such that $R^{i}f_*\mathbb Q_p$ is a lisse sheaf over $U$?

2)Assume $Y=spec(k)$ and $k$ finitely generated. Then $R^{i}f_{*}\mathbb Q_p$ correspond to the representation $Gal(k^{sep}|k)\rightarrow GL(H^i(X_{k^{sep}}, \mathbb Q_p))$. Is it possible that this representation is trivial or almost trivial?

When specialized to abelian varieties the second question becomes:

2')Is it possible that the action of the absolute Galois group of $k$ on the $p$ Tate module is trivial or almost trivial?

Let $k$ a field with $char(k)=p>0$, separable closure $k^{sep}$ and $f:X\rightarrow Y$ a smooth projective morphism of smooth variety over $k$.

1)Is it true that there exists a (EDIT) dense open susbset $U$ of $Y$ such that $R^{i}f_*\mathbb Q_p$ is a lisse sheaf over $U$?

2)Assume $Y=spec(k)$ and $k$ finitely generated. Then $R^{i}f_{*}\mathbb Q_p$ correspond to the representation $Gal(k^{sep}|k)\rightarrow GL(H^i(X_{k^{sep}}, \mathbb Q_p))$. Is it possible that this representation is trivial or almost trivial (EDIT i.e it has finite image)?

When specialized to abelian varieties the second question becomes:

2')Is it possible that the action of the absolute Galois group of $k$ on the $p$ Tate module is trivial or almost trivial?