TITLE: Reference for $p$-adic Hodge theory with coefficients
QUESTION [5 upvotes]: Let $K$ be a $p$-adic field and $L$ be a finite or infinite extension (maybe algebraic ?) of $\mathbb{Q}_p$.

Is there a reference for $p$-Hodge theory for representations $\rho : Gal_K \rightarrow GL_V(L)$ ?

In particular:

Is $Rep_L(Gal_K)$ classified by $Mod_{L \otimes_{\mathbb{Q}_p} \mathcal{E}}(\phi,\Gamma)$?
What is the definitions of 'de Rham representations' and 'Hodge-Tate weights'?

REPLY [4 votes]: If you are looking at $L$-linear representations of $Gal_K$ with $L$ a finite extension of $Q_p$ then everything works, basically by forgetting the $L$-linearity and then bringing it back later on. A good place to read about some of this would be Breuil-Mezard's "Multiplicites modulaires et representations de GL2(Zp) et de Gal(Qp/Qp) en l = p" where they prove that "weakly admissible" is the same notion whether you remember the coefficients or not, which is not really obvious. Concerning HT weights, you get a set of weights for each embedding of $L$ into $\overline{Q}_p$. 
If $L$ is $\overline{Q}_p$, then somewhere in Breuil-Mezard they prove that things are actually defined over a finite extension of $Q_p$.