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.

  • $\begingroup$ What you attribute to Faltings is stated incorrectly: you need to either weaken the "crystalline" condition or assume stronger "good reduction" hypotheses on $X$. (Likewise, Fontaine's result was not for "general" abelian varieties, but rather those with good reduction, studied via their $p$-divisible groups over the valuation ring.) $\endgroup$
    – user28172
    Jan 23 '13 at 8:28
  • 1
    $\begingroup$ An extension to more general base rings than an $p$-adic field is shown in a paper by Jean-Pierre Wintenberger, ams.org/mathscinet-getitem?mr=1293978, to be found in the volume 223 of Astérisque, ``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). $\endgroup$
    – ACL
    Jan 23 '13 at 8:51
  • $\begingroup$ Correction : "For abelian varieties with good reduction, an extension..." $\endgroup$
    – ACL
    Jan 23 '13 at 8:51
  • $\begingroup$ In the case of abelian varieties with good reduction (or even p-divisible groups), Faltings give a direct proof (avoiding the machinery of almost etale extensions, close in spirit to Fontaine's proof) in his "Integral crystalline cohomology over very ramified rings" paper. $\endgroup$
    – anon
    Jan 23 '13 at 9:49

See chap. II.3 of Colmez's book (Asterisque 248).


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