ag.algebraic geometry – All Galois characters showing up in cohomology of one family of varieties

Fix a prime $p$.

Can we find a smooth proper map $Xto Y$ of $mathbb{Q}_p$-varieties such that any given representation $mathrm{Gal}(overline{mathbb{Q}_p}/mathbb{Q}_p)to mathrm{GL}_1(mathbb{F}_p)$ embeds in $oplus_{igeq 0} H^i_{mathrm{acute{e}t}}(X_ytimes overline{mathbb{Q}_p}, mathbb{F}_p)$ for some $yin Y(mathbb{Q}_p)$?