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I'm reading Illusie's survey on Crystalline cohomology, and I found him talking about those $p$-adic period rings like $B_{\text{dR}}, B_{\text{cris}}$. Can anybody explain what they are and give some heuristic on why they are what they are and what they are good for? So far I've only seen them appear in comparison isomorphisms. Are they just there to make the comparisons work?

I think I found a related question here: Fontaine's rings of periods but answers there simply didn't say much about it.

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You should ask your advisor. He can give you a better answer than most people in the world. –  Minhyong Kim Mar 15 '12 at 0:33
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Allow me a few more didactic comments: A short answer to the question about comparison isomorphisms is 'yes,' but a longer answer will eventually reveal many more connections, e.g., to the theory of Galois representations, automorphic forms, etc. So go talk to Martin, figure things out a bit, and then, in the public interest as well as for your own education, post an answer below to your own question. This isn't to discourage others from answering, mind you! –  Minhyong Kim Mar 15 '12 at 1:19

1 Answer 1

Here are two examples where these period rings play a crucial rôle. They represent the point of view of a spectator. It is to be hoped that some of the actual players --- many of whom have enriched MO --- will chime in with their favourite examples.

$B_{dR}$.

Cuspidal eigenforms $f$ (of some level and weight) give rise to galoisian representations $\rho_{f,p}:\mathrm{Gal}(\overline{\mathbf{Q}}|\mathbf{Q})\to\mathrm{GL}_2(\overline{\mathbf{Q}}_p)$

for each prime $p$ (Shimura, Deligne, Serre). Such representations are called modular.

They are unramified away from finitely many primes (those which divide the level) and, crucially, derahmian (=$B_{dR}$-admissible) at $p$.

It is natural to ask : Which galoisian representations $\rho:\mathrm{Gal}(\overline{\mathbf{Q}}|\mathbf{Q})\to\mathrm{GL}_2(\overline{\mathbf{Q}}_p)$ arise from some cuspidal eigenform ?

Fontaine and Mazur conjectured that the necessary conditions for modularity enumerated above (along with other obvious ones) are also sufficient. This is now almost a theorem (Kisin-Emerton); see the recent Bourbaki talk by Laurent Berger.

$B_{cris}$.

Abelian varieties $A$ of dimension $g$ over a finite extension $K$ of $\mathbf{Q}_l$ give rise to galoisian representations

$\rho_{A,p}:\mathrm{Gal}(\overline{K}|K)\to\mathrm{GL}_{2g}(\mathbf{Q}_p)$

coming from the galoisian action on the $p$-power torsion points of $A$ (Weil, Tate). Does $\rho_{A,p}$ tell us whether $A$ has good reduction or not ?

The Néron-Ogg-Shafarevich theorem says that if $l\neq p$, then $A$ has good reduction if and only if the representation $\rho_{A,p}$ is unramified (see the Serre-Tate paper in the Annals).

What happens if $l=p$ ? Fontaine proved in this case that if $A$ has good reduction, then the representation $\rho_{A,p}$ is crystalline ($=B_{cris}$-admissible). Conversely, Coleman and Iovita have proved that if $\rho_{A,p}$ is crystalline, then $A$ has good reduction.

These are of course only two of the many things for which $B_{dR}$ (resp. $B_{cris}$) are essential.

Addendum An expert (who wants to remain anonymous) has pointed out to me that another proof of the implication "$\rho_{A,p}$ is crystalline $\Longrightarrow$ $A$ has good reduction" (in the case $l=p$) can now be given by combining an old result

(i) (Grothendieck, SGA7) If $\rho_{A,p}$ comes from a $p$-divisible group, then $A$ has good reduction,

with a conjecture of Fontaine as proved by

(ii) (Breuil, Annals 2000 for $p\neq2$, Kisin, Durham symposium 2007 for $p=2$) If $\rho_{A,p}$ is crystalline, then it comes from a $p$-divisible group.

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These are very good examples! Another example is that these rings of periods can be used to define the $H^1_f$ groups "at p". –  Laurent Berger Mar 17 '12 at 11:52

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