To identify $p$-adic Tate module $T_p(G)$ of $p$-divisible group $G$ in the category $\text{Rep}_{\mathbb{Q}_p}(G_{K_\infty})$

Question

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.

Answer 1

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).