Semistable Galois representation

Definition

Let $K/\mathbb Q_p$ be finite, let $K_0$ be its maximal unramified subfield, 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 $B_{\mathrm{st}}$ , set

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

The representation $V$ is semistable when

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

The resulting module carries Frobenius $\varphi$ , a nilpotent monodromy operator $N$ satisfying $N\varphi=p\varphi N$ , and a filtration after extension to $K$ . It is therefore a filtered $(\varphi,N)$ -module.

Potential semistability

The representation is potentially semistable if its restriction to $G_L$ is semistable for some finite extension $L/K$ . The $p$ -adic monodromy theorem says that this is equivalent to being de Rham.

A semistable representation is crystalline exactly when its monodromy operator vanishes. Geometrically, semistable reduction can produce nonzero monodromy, while good reduction produces the crystalline case.

Weil–Deligne representation

Fontaine's construction associates a Weil–Deligne representation to a potentially semistable representation. This is the object compared with the local Langlands parameter at a place above $p$ in precise formulations of local–global compatibility.

References

Jean-Marc Fontaine, “Représentations $p$ -adiques semi-stables,” Astérisque 223 (1994), 113–184. Numdam.
Laurent Berger, “Représentations $p$ -adiques et équations différentielles,” Inventiones Mathematicae 148 (2002), 219–284. arXiv.