Reference for $p$-adic Hodge theory with coefficients

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'?

• A quick and dirty way to answer the first part is to rephrase both sides as objects in the appropriate categories equipped with an action of L. – Keerthi Madapusi Pera Apr 11 '14 at 17:59
• Do you just mean representations over $L$ instead of $\mathbf{Q}_p$? Also, what is $\mathrm{GL}_V(L)$? Do you mean $V$ is a finite-dimensional $L$-vector space, and the group is $\mathrm{GL}(V)$? In any case, the various properties, de Rham, semistable, etc., can be defined in terms of the underlying $\mathbf{Q}_p$-vector space of dimension $\dim_L(V)[L:\mathbf{Q}_p]$. Also, for a more intrinsic (but equivalent) characterization, see Proposition C.2.2 on page 55 of Rebecca Bellovin's paper here: arxiv.org/abs/1306.5685 – Keenan Kidwell Apr 16 '14 at 1:02

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 $Q_p$.