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Is there an example of smooth and proper scheme $X \to \mathrm{Spec}(\mathbb Z)$, and an integer $i$ such that $H^i(X, \mathbb Q)$ is not a 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:

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  • $\begingroup$ Doesn't the result of Fontaine need $p \neq q$? $\endgroup$
    – user19475
    Commented Sep 25, 2017 at 12:22
  • $\begingroup$ @TimoKeller: Oo, yes of course. $\endgroup$
    – user114562
    Commented Sep 25, 2017 at 12:28
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    $\begingroup$ I think I calculated once that under GRH, Fontaine's argument goes through for $p+q \leq 4$. So one needs dimension at least 5. However, there is no real hope I see of doing this in any dimension $<11$. One approach would be to try to resolve the singularities of the coarse moduli space of $\overline{\mathcal M}_{1,11}$. Taking the Hilbert scheme might be a good first step, as this resolves some singularities in coarse moduli spaces of quotient stacks. $\endgroup$
    – Will Sawin
    Commented Sep 25, 2017 at 14:51
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    $\begingroup$ This is a notorious open problem. As Will says, there are natural examples of smooth proper DM stacks with interesting cohomology (e.g. $\overline{\mathcal{M}_{g,n}}$). But there is no known scheme example. $\endgroup$ Commented Sep 25, 2017 at 15:17
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    $\begingroup$ Is it any different if you just ask that $X$ has good reduction everywhere, i.e. for every $p$ there exists a smooth proper model $\mathscr X_p \to \operatorname{Spec} \mathbf Z_{(p)}$ (but the model is allowed to depend on $p$)? $\endgroup$ Commented Jul 16, 2020 at 14:48

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