Is there an example of smooth and proper

scheme$X \to \mathrm{Spec}(\mathbb Z)$, and an integer $i$ such that $H^i(X, \mathbb Q)$ isnota Hodge structure of Tate type?

Alternatively: such that $H^i(X_{\bar{\mathbb{Q}}}, \mathbb{Q}_\ell)$ is not a Galois representation of Tate type?

- A result of Fontaine says that $H^q(X,\Omega^p) = 0$ if $p \ne q$, and $p+q \le 3$. So we need $\dim(X) > 3$.
- If we allow stacks, then examples come from the theory of modular forms: $H^{11}(\bar{M}_{1,11})$ is associated with the Ramanujan $\Delta$ function. So this question is explicitly about
*schemes*.

An explicit example would be wonderful. An inexplicit proof that such an $X$ exists is fine as well.

Related questions:

stackswith interesting cohomology (e.g. $\overline{\mathcal{M}_{g,n}}$). But there is no known scheme example. $\endgroup$5more comments