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

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.

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).