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I've been trying lately to understand Fontaine's rings of periods, $B_{\mathrm{dR}}$, $B_{\mathrm{cris}}$, etc. However, I have a really hard time understanding and appreciating how to think about and use these. These rings seems so incredibly complicated and unintuitive it boggles my mind; I never seem to be able to remember their construction.

So, how do I think about, learn and use these obscure objects? (Do I really need to know all the details in their construction to appreciate and use them?)

In addition, I know the context and Fontaine's original motivation for considering these rings, but have they found any unexpected uses outside their intended domain?

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When you use the complex numbers and familiar facts such as contour integration, do you constantly think of the complexes as pairs $(x,y)$ where $x$ and $y$ are reals---which are equivalence classes of Cauchy sequences of rationals---which are equivalence classes of integers---which are equivalence classes of positive integers? No! You don't carry all that baggage around, you just learn and use standard facts about the complexes. I thoroughly recommend completely forgetting the definitions of these period rings at first, and instead learning the basic theorems about them e.g. comparison isos. –  Kevin Buzzard Apr 20 '11 at 20:26
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I agree with Kevin Buzzard. I've yet to find myself worried about their innards. To me these rings are a black box into which you place a Galois representation, and voila! out pops a pretty linear algebraic object. In some cases, you can put the linear algebraic object back into the black box and it comes back out as the Galois representation. It's probably more useful to be familiar with this process. –  Keerthi Madapusi Pera Apr 20 '11 at 20:31
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Daniel: hopefully others have answered the reference question. I absolutely understand where you're coming from. I remember watching lectures by Fontaine about 17 years ago where he spent 45 minutes writing down definitions and then a very rushed 10 minutes announcing theorems, and at the end of it I just thought "what was that all about??". All that has happened to me over the years is that I am in some sense still no closer to understanding what $B_{cris}$ is, but am a lot less scared of it simply because I've seen standard assertions about it made so often. –  Kevin Buzzard Apr 21 '11 at 7:51
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Here's the exercise you should perhaps do though---it certainly helped me. Find in the literature a statement of the classical de Rham isomorphism for the cohomology of a smooth projective algebraic variety over the complexes. Now say the variety was defined over the reals, and figure out how complex conjugation matches up on both sides (warning: the answer is not the first thing you'd guess). Now change the reals to the $p$-adics, change the complexes to $B_{dR}$ and observe that the comparison isomorphism says a completely analogous thing! Now you see the point of $B_{dR}$... –  Kevin Buzzard Apr 21 '11 at 7:55
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Here's something which may help: think of the elements of $Q_p^{alg}$ as functions of $p$. You can then think of the $p$-adic topology as "uniform convergence", and the completion of $Q_p^{alg}$ for this topology is $C_p$. Now think of the uniform convergence of functions along with the uniform convergence of their derivatives. This gives you a topology on $Q_p^{alg}$ and its completion is then $B_{dR}^+$. Forgetting this extra info gives the map $\theta : B_{dR}^+ \to C_p$. –  Laurent Berger Apr 21 '11 at 10:50

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I would say that Bloch-Kato conjecture (giving the exact value of complex $L$ functions of motives) was a rather unexpected application.

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I say! That's indeed a remarkable application. –  Daniel Larsson Apr 21 '11 at 9:25

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