notion of $\mathrm{Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)$ representation with complex multiplication

Question

In usual Hodge theory, there is the notion of Hodge structure $H$ with complex multiplication, that can be defined in several ways, i.e. asking that there exists a CM number field $E$ such that $\dim H=[E: \mathbb{Q}]$ and an embedding of $E$ into the endomorphisms of Hodge structures of $H$.

Is there a similar notion in $p$-adic Hodge theory, that is, a definition of $\mathrm{Gal}(\overline{\mathbb{Q}}_p/ \mathbb{Q}_p)$-representation with complex multiplication?

The easiest example to the notion should apply is the following: $E$ is a CM elliptic curve over $\mathbb{Q}$ and one considers $H^1_{dR}(E / \mathbb{Q}_p)$.

Answer 1

I think you want to consider virtually abelian representations - i.e., representations that factor through some virtually abelian group.

CM abelian varieties are exactly those whose Galois representations are virtually abelian as a representation of the full Galois group of $\mathbb Q$. So they will still be abelian as representations of the Galois group of $\mathbb Q_p$. But many non-CM abelian varieties will be CM in this sense as well. This is to be expected as p-adic Hodge theory carries much less information than usual Hodge theory.