Crystalline Galois representation

Definition

Let $K/\mathbb Q_p$ be finite, let $K_0$ be the maximal unramified subfield of $K$ , and let $V$ be a finite-dimensional $\mathbb Q_p$ -representation of the absolute Galois group $G_K=\operatorname{Gal}(\overline K/K)$ . Using the Fontaine period ring $B_{\mathrm{cris}}$ , set

D_{\mathrm{cris}}(V)= (B_{\mathrm{cris}}\otimes_{\mathbb Q_p}V)^{G_K}.

The representation $V$ is crystalline when

\dim_{K_0}D_{\mathrm{cris}}(V)=\dim_{\mathbb Q_p}V.

The period module has a semilinear Frobenius, and after extension from $K_0$ to $K$ it has the filtration inherited from $B_{\mathrm{dR}}$ . These data form a filtered Frobenius module.

Relation to reduction and monodromy

Crystalline representations are semistable with monodromy operator $N=0$ , hence are de Rham. The word “crystalline” is not synonymous with “unramified”: a crystalline representation can have nontrivial inertia action, although its ramification is tightly constrained.

For a smooth proper variety with good reduction over $K$ , $p$ -adic étale cohomology is crystalline and compares with crystalline cohomology of the special fiber. This good-reduction paradigm motivates the terminology.

References

Jean-Marc Fontaine, “Le corps des périodes $p$ -adiques,” Astérisque 223 (1994), 59–111. Numdam.
Olivier Brinon and Brian Conrad, CMI Summer School Notes on $p$ -adic Hodge Theory, 2009, Chapters 9–10. Author notes.