If $X$ is a smooth projective variety over a $p$-adic field $K$, then Faltings' Theorem says that the etale cohomology of $X_{\overline{K}}$ is crystalline.

There have been various steps towards this result. Tate showed that the cohomology of an abelian variety with good reduction is Hodge-Tate. Fontaine gave a somewhat "elementary" proof of this for general abelian varieties, and also showed that the cohomology is not only Hodge-Tate, but crystalline. That implies that it is de Rham, i.e. $B_{dR}$-admissible.

Is there a direct proof in the literature of the this fact ($B_{dR}$-admissibility) for abelian varieties? I would imagine that one should be able to give a simpler argument than what's required to show that it's crystalline.

223ofAstérisque, ``Périodes $p$-adiques'' (edited by Fontaine). I would bet than one can obtain the more general result along the same lines (at least if the ab. var. has semi-stable reduction over $K$) by adding as an input results of Raynaud in the same volume, or uses the universal vector extension of (the connected component of) a Néron model (Mazur-Messing). $\endgroup$