What is an example of a $p$-adic representation of the absolute Galois group of a $p$-adic field that has finite image on the inertia subgroup, but is not crystalline?



Finite image but not crystalline
If a crystalline representation has finite image when restricted to inertia, then this restriction has to be trivial. 


Indeed, suppose that $K$ is a discretely valued extension of $\mathbb{Q}\_p$ and $\rho:G\_K\to GL(V)$ is a crystalline representation which has finite image when restricted to the inertia subgroup. Replace $K$ by the completion $K^{nr}$ of its maximal unramified extension to get a crystalline representation $\rho: G\_{K^{nr}}\to GL(V)$ with finite image. Let $L\supset K^{nr}$ be the finite extension corresponding to the kernel of $\rho$.


Then $(V\otimes\_{\mathbb{Q}\_p} B\_{cris})^{G\_{K^{nr}}}=((V\otimes B\_{cris})^{Gal(\bar{K}/L)})^{Gal(L/K^{nr})}=(V\otimes K^{nr})^{Gal(L/K^{nr})}$ where in the last tensor product $Gal(L/K^{nr})$ acts trivially on $K^{nr}$ and via the representation $\rho$ on $V$. The last equality used the



> 
> **Lemma.**(Proposition 5.1.2 in Fontaine's "Repr'esentations p-adiques semi-stables" or Theorem 9.2.10 in Brinon and Conrad's notes) If $L/K$
>  is a finite extension then
>  $(B\_{cris})^{Gal(\bar{K}/L)}=L\_0$ where $L\_0\subset L$ is
>  the maximal unramified extension of $\mathbb{Q}\_p$ contained in $L$.
> 
> 
> 


Thus, $\dim\_{K^{nr}}(V\otimes B\_{cris})^{G\_{K^{nr}}}<\dim\_{\mathbb{Q}\_p}V$ unless $\rho:Gal(L/K^{nr})\to GL(V)$ is trivial.


An example of representation with finite but nontrivial image of inertia exists already in dimension $1$ if $K$ does not contain a $p$-th root of unity: take $\rho$ to be $G\_K\to Gal(K(\mu\_p)/K)\simeq (\mathbb{Z}/p)^{\times}\xrightarrow{[\cdot]}\mathbb{Q}\_p^{\times}$ where $[\cdot]$ denotes the Teichmuller representative. If $K$ contains a $p$-th root of unity then we can take $\rho:G\_K\to Gal(K(\mu\_{p^n})/K)\simeq \mathbb{Z}/p\xrightarrow{\text{regular represntation}} GL\_{p}(\mathbb{Q}\_p)$ where $n$ is the smallest integer such that $K$ does not contain a $p^n$-th root of unity.


