Absolutely irreducible p-adic representation of the absolute Galois group of Q_p

Question

Let $p$ be a prime number, $\mathbb{Q}_p$ the field of $p$-adic numbers, $G_p$ the absolute Galois group of $\mathbb{Q}_p$ and $V$ a finite dimensional vector space over $\mathbb{Q}_p$. Assume we are given a linear representation $\rho : G_p \to GL(V)$.

Can we find a closed subgroup of finite index $H$ in $G_{p}$ such that the restriction of $\rho$ to $H$ is reducible ?

It is possible in the case of a representation which have coefficients in $\mathbb{F}_p$ (in this case, absolutely irreducible representations are induced).

Answer 1

The answer is no. To prove it, it suffices to show that there exists a representation $\rho: G_p \rightarrow Gl(V)$ of open image. There are several ways to do this, one is to use Chenevier prop. 1.8 in "Quelques courbes de Hecke se plongent dans l'espace de Colmez", which says that for $p=2,3,5,7$ and $\rho: G_{\mathbb Q,p} \rightarrow Gl_2(\mathbb Q_p)$ an odd representation (where $G_{\mathbb Q,p}$ is the global galois group of $\mathbb Q$ unramified outside $p$), one has $\rho(G_{\mathbb Q,p})=\rho(G_p)$. So it suffices to find now a "global" representation with open image, and this is well-known to exist (e.g. Serre's open image for elliptic curve, etc.)

Answer 2

One gets loads of crystalline counterexamples using $p$-divisible groups (and more specifically from any elliptic curve with supersingular reduction).

Let $k$ be a perfect field of characteristic $p > 0$ and let $K$ be a complete discretely-valued field of characteristic 0 having residue field $k$. Let $\Gamma_0$ be a $p$-divisible group over $k$. By the unobstructedness of the infinitesimal deformation theory of $p$-divisible groups, this lifts to a $p$-divisible group $\Gamma$ over the valuation ring $O_K$ of $K$ (and even over $W(k)$). Suppose that the representation of the Galois group of $K$ associated to the generic fiber $\Gamma_K$ becomes reducible on the finite-index open subgroup, say associated to a finite extension $K'/K$. This is exactly the condition that $\Gamma_{K'}$ admits a nontrivial filtration, and by the usual schematic closure trick (and the work of Raynaud/Tate on $p$-divisible groups) that uniquely extends to such a filtration of $\Gamma_{O_{K'}}$, and hence of $(\Gamma_0)_{k'}$ for the residue field $k'/k$ of $K'/K$. In other words, as long as $\Gamma_0$ is a "geometrically isosimple" $p$-divisible group (i.e., $(\Gamma_0)_{\overline{k}}$ is simple in the isogeny category over $\overline{k}$) then all such $\Gamma_K$ furnish examples.

The Dieudonne-Manin classification provides lots of concrete Dieudonne modules corresponding to geometrically isosimple $\Gamma_0$ (and then by the Fontaine-Honda formalism or other formalisms one can "write down" the data of lots of lifts over $W(k)$). And in more concrete terms, the $p$-divisible group of any supersingular elliptic curve over $k$ is of this type (so the $p$-adic Tate module of any elliptic curve over $K$ with supersingular reduction does the job) since a nontrivial composition series would (by height reasons) have to consist of terms which are either multiplicative or etale, contradicting the supersingularity hypothesis.