Hodge–Tate representation

Definition

Let $K/\mathbb Q_p$ be a finite extension, let $G_K$ be its absolute Galois group, and let $V$ be a finite-dimensional $\mathbb Q_p$ -representation of $G_K$ . It is Hodge–Tate if there is a $G_K$ -equivariant decomposition

\mathbb C_p\otimes_{\mathbb Q_p}V \simeq\bigoplus_{i\in\mathbb Z} \mathbb C_p(-i)^{m_i}.

The integers $i$ , repeated with multiplicity $m_i$ , are the Hodge–Tate weights in the convention used here. Some authors attach weight $-i$ to $\mathbb Q_p(i)$ , so signs must be checked when comparing sources.

Period-ring criterion

With the Fontaine period ring $B_{\mathrm{HT}}=\bigoplus_{i\in\mathbb Z}\mathbb C_p(i)$ , set

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

Then $V$ is Hodge–Tate exactly when the total $K$ -dimension of this graded space equals $\dim_{\mathbb Q_p}V$ . The grading records the weights.

Every de Rham representation is Hodge–Tate, but not conversely. Hodge–Tate weights are the local $p$ -adic Hodge invariants that appear in the algebraicity conditions on automorphic Galois representations.

References

Olivier Brinon and Brian Conrad, CMI Summer School Notes on $p$ -adic Hodge Theory, 2009, §§2.3–2.4. Author notes.
Jean-Marc Fontaine, “Représentations $p$ -adiques des corps locaux. I,” in The Grothendieck Festschrift II, Progress in Mathematics 87, 1990.