Finite image but not crystalline

Question

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?

Answer 1

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.