De Rham Galois representation

Definition

Let $K/\mathbb Q_p$ be finite, let $G_K=\operatorname{Gal}(\overline K/K)$ be its absolute Galois group, and let $V$ be a finite-dimensional $\mathbb Q_p$ -representation of $G_K$ . Its space of de Rham periods, defined using the Fontaine period ring $B_{\mathrm{dR}}$ , is

D_{\mathrm{dR}}(V)= (B_{\mathrm{dR}}\otimes_{\mathbb Q_p}V)^{G_K},

a filtered $K$ -vector space. The representation $V$ is de Rham when

\dim_KD_{\mathrm{dR}}(V)=\dim_{\mathbb Q_p}V.

The decreasing filtration inherited from $B_{\mathrm{dR}}$ encodes the Hodge filtration. Its jumps recover the Hodge–Tate weights, up to the stated sign convention.

Place in p-adic Hodge theory

There are implications

\text{crystalline}\Longrightarrow\text{semistable} \Longrightarrow\text{de Rham}\Longrightarrow\text{Hodge–Tate}.

The $p$ -adic monodromy theorem strengthens the middle relation: a representation is de Rham if and only if it becomes semistable after a finite extension of $K$ . Thus “de Rham at places above $p$ ” is a robust local condition in automorphic constructions of Galois representations.

Geometric origin

If $X/K$ is smooth and proper, its $p$ -adic étale cohomology is de Rham; the comparison isomorphism identifies $D_{\mathrm{dR}}$ with algebraic de Rham cohomology. Smooth proper varieties with good reduction give representations satisfying the stronger crystalline condition.

References

Olivier Brinon and Brian Conrad, CMI Summer School Notes on $p$ -adic Hodge Theory, 2009, Chapters 4–6. Author notes.
Laurent Berger, “Représentations $p$ -adiques et équations différentielles,” Inventiones Mathematicae 148 (2002), 219–284. arXiv.