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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Apparently this issue was worked out shortly after, during Go Yamashita's stay at the IHES in 2006, and after some discussion with Deligne.
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 $M$.
This realization is functorial, and has the comparison isomorphism:
$$k_v \otimes_{k_0,v}D_\mathrm{crys}(M_p) \cong k_v\otimes_k M_\mathrm{dR}$$
All of this is perfectly explained and proved in section 4 of:
- Go Yamashita, Bounds for the Dimensions of p-adic Multiple L-Value Spaces (2009)
