# Crystalline realization of mixed Tate motives

Deligne and Goncharov, in their article of 2005, mention that the crystalline realization functor has yet to be worked out. What's the current state of the literature on this? And how big of an issue is it?

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For any Tate motive $M$ unramified at $v$, its crystalline realization is defined as $D_\mathrm{crys}(M_p)$, where $D_\mathrm{crys}$ is Fontaine's functor $(B_\mathrm{crys} \otimes_{\mathbb{Q}_p}-)^{G_{k_v}}$, and $M_p$ is the p-adic realization of $M$.
$$k_v \otimes_{k_0,v}D_\mathrm{crys}(M_p) \cong k_v\otimes_k M_\mathrm{dR}$$