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Let $X/K$ be a variety over a global field $K$. When (and why) is the Galois representation $H^i_{et}(X \times_K \bar{K}, \mathbf{Q}_\ell)$ unramified at a place $v$ of $K$?

I guess this is true if $X$ has a model smooth (or regular?) over $v$ by using base change theorems.

Same question for Hodge-Tate/de Rham/crystalline/semistable.

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  • $\begingroup$ You want $v$ to be a place over $\ell$, right? Otherwise the last part doesn't make sense. $\endgroup$
    – JBorger
    Mar 12 '10 at 21:38
  • $\begingroup$ Yes, except for unramified, thanks. $\endgroup$
    – user19475
    Mar 13 '10 at 6:44
  • $\begingroup$ Related: mathoverflow.net/questions/18006 $\endgroup$
    – Watson
    Sep 26 '18 at 17:52
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[Added: Just to set the scene, this answer is discussing the $p$-adic etale cohomology of a variety $X$ over a finite extension of $\mathbb Q_p$ (or, a little more generally, a finite extension of the fraction field of the Witt vectors of a perfect field of char. $p$).]

Crystalline implies semi-stable implies de Rham implies Hodge-Tate. The cohomology is crystalline if $X$ is proper with a smooth model, is semi-stable if $X$ is proper with a semi-stable model, and is de Rham always. (In the proper case, the last statement is a theorem of Tsuji; in the general case, of Kisin. The crystalline and semi-stable cases are due, in various degrees of generality, to Fontaine--Messing, Hyodo--Kato, Faltings, and Tsuji.)

Added: It might help to note that de Rham and potentially semi-stable coincide (as discussed in the comments below, Berger reduced this to a result in the theory of $p$-adic differential equations, which was then proved independently by Andre, Kedlaya, and Mebkhout). Also, unramified is a very strong condition in this setting ($p$-adic etale cohomology of varieties over $p$-adic fields), which essentially never holds unless we are looking just at $H^0$, and if the geometrically connected components of $X$ are defined over an unramified extension.

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  • $\begingroup$ I think it is more usual to attribute the equivalence of de Rham and potentially semi-stable to Berger (although there are now several proofs). $\endgroup$
    – TSG
    Mar 13 '10 at 20:59
  • $\begingroup$ Thanks; I edited the above. (Colmez and Fontaine proved a different conjecture of Fontaine!) $\endgroup$
    – Emerton
    Mar 14 '10 at 12:22
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    $\begingroup$ The equivalence `` de Rham iff potentially semistable '' is due independently to André, Kedlaya and Mebkhout; all three use Berger's results. Here is what Fontaine (arxiv.org/abs/math/0304232) writes : Berger en a ramené la preuve à un résultat sur les équations différentielles p-adiques, résultat qui a ensuite été prouvé indépendamment par André, Kedlaya et Mebkhout. $\endgroup$ Mar 15 '10 at 3:56
  • $\begingroup$ Dear Chandan, Thanks for your comment. I had forgotten the order of events (whether Berger's result came before or after the work on $p$-adic differential equations of Andre, Kedlaya, and Mebkhout), but no doubt Fontaine has gotten it right. It is probably best to mention all four authors, and I will edit my post to reflect this. $\endgroup$
    – Emerton
    Mar 15 '10 at 22:48
  • $\begingroup$ the most mysterious thing for me is how they actually proved these things. $\endgroup$
    – natura
    Mar 24 '10 at 13:23
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By asking "when is it ramified," you might be also be asking whether there are any other conditions under which it's ramified, i.e. a converse to Emerton's answer. For abelian varieties, the converse is true, and this is known as N\'eron-Ogg-Shafarevich. There are $p$-adic Hodge theory versions as well.

However, in general, a variety with Galois representation unramified at $v$ need not have good reduction at $v$. In all cases where this has been proven, one relates the Galois representation in question to the representations of auxiliary varieties that do have good reduction at $v$. As a simple example, there are smooth projective curves with bad reduction at $v$ but whose Jacobian has good reduction at $v$.

One might conjecture that any Galois representation coming from a variety actually comes in some fashion from varieties with good reduction over $v$. More precisely, any motive over a global field $K$ whose associated $\ell$-adic Galois representation is unramified at a finite place $v$ (whose residue characteristic $p$ is not $\ell$) is the base change of a motive (of smooth proper varieties) over $\mathcal{O}_{K,v}$, the local ring at $v$. One can make a similar conjecture for $\ell=p$ if one assumes the representation is crystalline. One can even make a modification by defining a category of motives using semistable varieties. I don't know if this is written down anywhere, but I and other graduate students I know have wondered this.

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As far as I remember, the representation is always potentially semistable (this means that it is Hodge-Tate and de Rham?). It is crystalline when $X$ has a smooth model; it is semistable if it has a log-smooth model (I am not sure that this is the right name).

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The question about unramified is still open. (Sorry for using the answer for bumping up.)

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    $\begingroup$ Again, are you talking about when $v$ divides $\ell$ or when it doesn't? They're completely different. In the first case, Emerton gave an answer above. In the second case, it happens if the variety has a smooth proper model over $v$, by the proper base change theorem. $\endgroup$
    – JBorger
    Mar 18 '10 at 21:48
  • $\begingroup$ I want the case $\nu \nmid \ell$. I don't see how unramifiedness follows from the proper base change theorem. $\endgroup$
    – user19475
    Mar 19 '10 at 8:00
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    $\begingroup$ Well, that's a long story and one which I haven't studied well. The result is stated in SGA 4 1/2 [Arcata] V (3.1) (on page 62). So you can start reading there! $\endgroup$
    – JBorger
    Mar 21 '10 at 9:41
  • $\begingroup$ If you want to bump up, you can always edit the original post. $\endgroup$ Nov 1 '12 at 23:12

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