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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:

user114562
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  • Doesn't the result of Fontaine need $p \neq q$? –  Sep 25 '17 at 12:22
  • @TimoKeller: Oo, yes of course. – user114562 Sep 25 '17 at 12:28
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    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. – Will Sawin Sep 25 '17 at 14:51
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    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. – Daniel Litt Sep 25 '17 at 15:17
  • @DanielLitt: Yes, I am aware of those natural examples (see the question, where I mention $\overline{M}_{1,11}$). One line of attack that I was thinking about for a proof of existence was to use general results about categories of motives built from smooth projective varieties versus motives built from smooth proper DM stacks. I think that over fields there are theorem that say they give the same category. I don't know if something like this also works over $\mathbb Z$. – user114562 Sep 25 '17 at 17:01
  • @WillSawin: Ok, that is good to know. I don't expect explicit examples with dimension $< 11$. But even an explicit example with dimension $\ge 11$ does not seem trivial. – user114562 Sep 25 '17 at 17:03
  • @user114562 I don't think this could work because the reduction passes through possibly singular schemes, and you need alteration or resolution to get back to smooth schemes, neither of which work over $\mathbb Z$. Essentially your problem is just a big Diophantine equation, and there is very rarely a nonconstructive existence proof for solutions to a Diophantine equation. – Will Sawin Sep 26 '17 at 10:34
  • Chenevier and Lannes have pursued the automorphic approach (initiated by Mestre and Fermigier) to this question, generalizing the Weil explicit formula. Namely, they classify the algebraic cuspidal representations of $\mathrm{PGL}_n$ of motivic weight $\leq 22$ and show that the belong to an explicit list of $11$ representations. In their list, only the trivial representation has motivic weight $<11$ and corresponds to Ramanujan's $\Delta$. – ACL Sep 26 '17 at 19:48
  • @ACL: Right, so that suggest that there definitely shouldn't be anything with dim $< 11$. Do some Langlands-type conjectures imply that there should be smooth proper schemes over $\mathbb Z$ with non-Tate cohomology? Or is it expected that we need to include DM-stacks to get all the motives that appear in the Chenevier-Lannes list? – user114562 Sep 27 '17 at 06:10
  • 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 $p$)? – R. van Dobben de Bruyn Jul 16 '20 at 14:48

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