# Fontaine's classification of p-divisible groups

Let k be a finite extension of $\mathbb{F}_p$, and W its ring of Witt vectors. Write W[F,V] for the Dieudonn\'e ring.

Let G be a connected p-divisible group which is finite-dimensional over k, and let M be its contravariant Dieudonne module. Define $G(k[[T]])=\varprojlim G(k[[T]]/(T^n))$. Then Fontaine's classification theorem of p-divisible groups implies that there is a canonical isomorphism $\theta:G(k[[T]]) \cong {\rm Hom}_{W[F,V]}(M,\widehat{{\rm CW}}(k[[T]])),$ where $\widehat{{\rm CW}}(k[[T]])$ is the completed ring of Witt convectors of k[[T]]. Can one give an explicit description of this isomorphism?

In her paper "Theorie d'Iwasawa globale and locale", Perrin-Riou considers the 'submodule of logarithms' L of M, which has the property that ${\rm Fil}^1M=VL$. Let H be the subgroup of G(k[[T]]) whose image under $\theta$ factors through M/L. Can one describe H without using the isomorphism $\theta$?

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I guess this is given by the "evaluation" map: By definition, $M=\text{Hom}_{W[F,V]}(G,CW)$ and if you have an element $g$ in $G(k[[t]])$, evaluation at $g$ will induce a map from $M$ to $CW(k[[t]])$.