# 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 $$yin Y(mathbb{Q}_p)$$?