The image of the Galois group for some crystalline representations

Let $K$ be the field of fractions of the ring $W=W(k)$ of Witt vectors, where $k$ is an algebraically closed field of characteristic $p>0$, and let $\Gamma=\operatorname{Gal}(\overline K/K)$. If $U$ is a $\Gamma$-invariant lattice in a continuous $\mathbb Q_p[\Gamma]$-module $V$ of finite dimension over $\mathbb Q_p$ and if the set of characters $S$ of the semisimple envelope of $U\otimes\mathbb F_p$ satisfies some additional assumptions, then one can associate to $U$ a function $n_U\colon S\times S\to\mathbb Z_{\geqslant 0}\cup\{\infty\}$ containing a considerable amount of information about the image $H_U$ of $\Gamma$ in $\operatorname{Aut}_{\mathbb Z_p}U$. In this paper we describe the set of functions arising from crystalline modules $V$ with Hodge–Tate weights in the interval $[0,p-2]$. Moreover, we explicitly express these functions in terms of the corresponding filtered modules. This is applied to the description of the image $H_{T(\mathcal G)}$, where $T(\mathcal G)$ is the Tate module of a 1-dimensional formal group $\mathcal G$ over $W(k)$ of finite height.