# 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. Apr 11, 2014 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 Apr 16, 2014 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 $Q_p$.