TITLE: Fontaine's rings of periods
QUESTION [20 upvotes]: 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?

REPLY [7 votes]: I would say that Bloch-Kato conjecture (giving the exact value of complex $L$ functions of motives) was a rather unexpected application.