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

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