Let $k$ be a perfect field of characteristic $p>0$, $W=W(k)$ its ring of Witt vectors, $K\_0=W(k)[\frac{1}{p}]$ and, $K/K\_0$ be a totally ramified extension. Let $\pi \in K$ be an uniformizer.


Consider the Kummer/Breuil extension $K\_{\infty}:=\bigcup\_{n \geq 0} K\_n$, where $K\_n:=K(\pi\_n)$ and define $G\_{K\_{\infty}}=\text{Gal}(\bar K/K\_{\infty}) \subset G\_K=\text{Gal}(\bar K/K)$, where $\bar K$ is the algebraic closure of $K$.


**My question:**


Denote by $\text{Rep}\_{\mathbb{Q}\_p}(G\_{K\_\infty})$ the category of continuous representations of $G\_{K\_\infty}$ on finite dimensional $\mathbb{Q}\_p$-vector spaces or in other words, $\text{Rep}\_{\mathbb{Q}\_p}(G\_{K\_\infty})$ is the category of $p$-adic representations of $G\_{K\_\infty}$. Also denote by $\text{Rep}\_{\mathbb{Z}\_p}(G\_{K\_\infty})$ is the category of $\mathbb{Z}\_p$-representations. Now $p$-adic Tate module $T\_p(G)$ of $p$-divisible group $G$ is equipped with structure of $\mathbb{Z}\_p$-module.
\begin{align}
&\text{(1) When $T\_p(G) \in \text{Rep}\_{\mathbb{Q}\_p}(G\_{K\_\infty}) \ \text{or} \ \text{Rep}\_{\mathbb{Z}\_p}(G\_{K\_\infty})$ ?} \\
&\text{(2) What is the relation between the category $\text{Rep}\_{\mathbb{Q}\_p}(G\_{K\_\infty})$ and the category } \\ & \hspace{2cm} \text{ of $p$-adic Tate modules $T\_p(G)$ of $p$-divisible groups $G$ ?} \\
& \text{(3) Is the functor $\text{Rep}\_{\mathbb{Z}\_p \ or \ \mathbb{Q}\_p}(G\_{K\_{\infty}})$} \longrightarrow \{\text{category of} \ T\_p(G) \} \ full \ or \ surjective ? 
\end{align}


I am trying to identify $T\_p(G)$ in the category $\text{Rep}\_{\mathbb{Q}\_p}(G\_{K\_\infty})$ or in $\text{Rep}\_{\mathbb{Z}\_p}(G\_{K\_\infty})$.


**Case I:** When we consider $G$ is just the abelian group $\text{of roots of unity}$ in the separable closure $K^s$ of $K$, then the $p$-adic Tate module is the rank one free $\mathbb{Z}\_p$-module equipped with action of the absolute Galois group $G\_K$ of $K$. That is, in this case the $p$-adic Tate module is the Galois representation or $p$-adic cyclotomic character of $K$.


**Case II:** When $G$ is the abelian variety over the field $K$, the $K^s$-valued points of $G$ form an abelian group. The $p$-adic Tate module $T\_p(G)$ of $G$ is the Galois representation of the absolute Galois group $G\_K$ of $K$.


But,more generally, I am trying to identify $T\_p(G)$ in the category $\text{Rep}\_{\mathbb{Q}\_p}(G\_{K\_\infty})$ or in $\text{Rep}\_{\mathbb{Z}\_p}(G\_{K\_\infty})$.


Any help would be appreciated.



To identify $p$-adic Tate module $T_p(G)$ of $p$-divisible group $G$ in the category $\text{Rep}_{\mathbb{Q}_p}(G_{K_\infty})$
I think Kisin's article "Crystalline representations and $F$-crystals" may be helpful. If you haven't read this, the following is a quick mention of some results.


There are several ways to express $T\_pG$ in the form of a $\mathbf{Z}\_p[G\_{K\_{\infty}}]$-module.
For example, using Breuil-Kisin modules (cf. Corollary 2.1.4; Lemma 2.2.4; Theorem 2.2.7).   

In practice, it is difficult to express $T\_pG$ in the form of a $\mathbf{Z}\_p[G\_K]$-module by Breuil-Kisin modules (notice that $\mathfrak{S}$ is only stable under $G\_{K\_{\infty}}$ not under $G\_{K}$; the expression in Corollary 2.1.4 hence has no obvious $G\_K$-action, but a $G\_{K\_{\infty}}$-action), that's something not so good about Breuil-Kisin module in my opinion.


We have the forgetful functor $\mathbf{Rep}\_{\mathbf{Z}\_p}(G\_K)\to \mathbf{Rep}\_{\mathbf{Z}\_p}(G\_{K\_{\infty}})$, a good news is that this functor is fully faithful when we restrict on the subcategory of crystalline representations $\mathbf{Rep}\_{\mathbf{Z}\_p}^{cris}(G\_K)$ (cf. Corollary 2.1.14).


