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Suppose $M$ is an object of the (conjectured) abelian category of mixed motives over $\mathbb{Q}$, $\textbf{MM}_{\mathbb{Q}}$, Scholl defines that $M$ is defined over $\mathbb{Z}$ if the following condition is satisfied.

Let $M_{\ell}$ be the $\ell$-adic realisation of $M$, which has a weight filtration $W_*$. For every $ p \neq \ell$, when we consider $M_{\ell}$ as a representation of the inertia group $I_p$, the weight filtration $W_*$ splits.

Suppose $X$ is a hypersurface of the projective space $\mathbb{P}^n_{\mathbb{Z}}$, and let $h^i(X)$ be the mixed motive associated to its $i$-th cohomology, is $h^i(X)$ (expected to be) a mixed motive defined over $\mathbb{Z}$?

If we further assume the singularity of $Y$ consists of finitely many $\mathbb{Z}$-valued points, then is $h^i(X)$ (expected to be) a mixed motive defined over $\mathbb{Z}$?

Here let's assume the existence of the category of mixed motives and its expected properties!

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In your first question, what do you mean by $h^i(X)$? Assuming you mean $h^i(X_\mathbb{Q})$, then its $\ell$-adic realization is the étale cohomology of $X_{\overline{\mathbb{Q}}}$, which is described (after restriction to a decomposition group at $p$) by a spectral sequence involving the étale cohomology of the special fiber of $X$ at $p$. It is certainly not true in general that the weight filtration splits even after retricting to inertia (see e.g. the weight-monodromy conjecture). On the other hand if you assume $X_{\mathbb{Q}}$ to be smooth projective over $\mathbb{Q}$ then $h^i(X)$ is defined over $\mathbb{Z}$ because the motive is pure of weight $i$, so the weight filtration Scholl refers to is trivial.

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  • $\begingroup$ Yes I mean $h^i(X_{\mathbb{Q}})$. For a smooth variety $X$ over $\mathbb{Q}$, $h^i(X_{\mathbb{Q}})$ is pure of weight $i$, so from the definition it is defined over $\mathbb{Z}$. If $X$ has very mild singularity, e.g. only one rational singular point, let's say it has weight filtration $W_{i-1} \subset W_i$, do you think in general it does not split into $W_{i-1} \oplus W_i/W_{i-1}$ considered as representation of inertia group? Do you know any references that might be helpful? $\endgroup$ – Wenzhe Apr 4 '18 at 14:13
  • $\begingroup$ If $E$ is an elliptic curve over $\mathbb{Q}_p$ with split multiplicative reduction then its Tate module sits in an exact sequence $0 \to \mathbb{Q}_\ell(1) \to V_\ell(E) \to \mathbb{Q}_\ell \to 0$ so there is a subspace of weight -2 and a quotient of weight 0. The action of inertia is trivial on $\mathbb{Q}_\ell(1)$ and $\mathbb{Q}_\ell$, but is unipotent on $V_\ell(E)$, hence the action of inertia does not split. $\endgroup$ – François Brunault Apr 4 '18 at 15:53
  • $\begingroup$ The following lecture notes by V. Dochitser could be helpful to you: arxiv.org/pdf/1410.1039.pdf $\endgroup$ – François Brunault Apr 4 '18 at 15:55
  • $\begingroup$ Feel sorry for bothering your again with a naive question. In section 4.5 of Nekovar's note Beilinson's conjecture (also in Scholl's note Remarks on special values of $L$-functions), it says that the category of pure motives form a full subcategory of category of mixed motives defined over $\mathbb{Z}$, does this mean every pure motive is defined over $\mathbb{Z}$? But the elliptic curve counter-example is against this statement. (I feel there is something I have not understood.) $\endgroup$ – Wenzhe Apr 4 '18 at 18:22
  • $\begingroup$ @Wenzhe You're welcome. I have to say I'm a bit confused by the definition of Scholl. In Groupes fondamentaux motiviques de Tate mixte, Deligne and Goncharov define mixed Tate motives over $\mathbb{Z}$ as those mixed Tate motives over $\mathbb{Q}$ which are unramified, in the sense that the associated $\ell$-adic representation is unramified at all primes $\neq \ell$ (see Proposition 1.7). It's not clear to me what is the relation between these two definitions. $\endgroup$ – François Brunault Apr 4 '18 at 18:22

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