Let $p$ be a prime number $\geq 3$. Let $V$ be a representation of $Gal(\bar{\mathbb{Q}}_p/ \mathbb{Q}_p)$ with coefficients in $\mathbb{F}_p$. Assume $V$ is a non-split extension of two characters $\delta_1$ and $\delta_2$.

Let $D$ be the $(\varphi, \Gamma)$-module associated to $V$. It is a non-split extension of two $(\varphi, \Gamma)$ of rank $1$. Denote by $\Phi$ the matrix of $\varphi$ and by $G$ the matrix of a generator $\gamma$ of $\mathbb{Z}_p^*$. In a suitable basis, we have $\Phi = \begin{pmatrix} f_1 & f \\ 0 & f_2 \end{pmatrix}$ and $G = \begin{pmatrix} g_1 & g \\ 0 & g_2 \end{pmatrix}$ where $f_1, f_2, g_1$ and $g_2$ are elements of $\mathbb{F}_p^{\times}$ (see One dimensional (phi,Gamma)-modules in char p) and where $f,g \in \mathbb{F}_p ((X))$.

Can we find a basis where $f$ and $g$ are in $\mathbb{F}_p[[X]]$ ? If not, what is the "nicest" form possible for $f$ and $g$ ? (i.e. can we kill some of the denominators and which one ?)

Let $p$ be a prime number $\geq 3$. Let $V$ be a representation of $Gal(\bar{\mathbb{Q}}_p/ \mathbb{Q}_p)$ with coefficients in $\mathbb{F}_p$. Assume $V$ is a non-split extension of two characters $\delta_1$ and $\delta_2$.

Let $D$ be the $(\varphi, \Gamma)$-module associated to $V$. It is a non-split extension of two $(\varphi, \Gamma)$ of rank $1$. Denote by $\Phi$ the matrix of $\varphi$ and by $G$ the matrix of a generator $\gamma$ of $\mathbb{Z}_p^*$. In a suitable basis, we have $\Phi = \begin{pmatrix} f_1 & f \\ 0 & f_2 \end{pmatrix}$ and $G = \begin{pmatrix} g_1 & g \\ 0 & g_2 \end{pmatrix}$ where $f_1, f_2, g_1$ and $g_2$ are elements of $\mathbb{F}_p^{\times}$ (see One dimensional (phi,Gamma)-modules in char p) and where $f,g \in \mathbb{F}_p ((X))$.

Can we find a basis where $f$ and $g$ are in $\mathbb{F}_p[[X]]$ ? If not, what is the "nicest" form possible for $f$ and $g$ ? (i.e. can we kill some of the denominators and which one ?)