7
$\begingroup$

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.

$\endgroup$
4
  • $\begingroup$ 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, 2013 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, 2013 at 8:51
  • $\begingroup$ Correction : "For abelian varieties with good reduction, an extension..." $\endgroup$
    – ACL
    Jan 23, 2013 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, 2013 at 9:49

1 Answer 1

1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.