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.