Trianguline representation

Question

I have a problem in understanding the concept of trianguline representation. Maybe someone can enlighten me.

Let $K$ be a finite extension of $\mathbb{Q}_p$ and $V$ be a $p$-adic representation of $G_K$ (absolute Galois group of $K$) of dimension $n$. By $p$-adic Hodge theory $V$ corresponds to some étale $(\varphi,\Gamma)$-module $M$ over the Robba ring $\mathcal{R}_K$. Lets call $M$ trianguline if there exists a filtration $$ 0=M_0 \subset M_1 \subset \cdots \subset M_n=M,$$ of sub-$(\varphi,\Gamma)$-modules of $M$ such that the successive quotients $M_i/M_{i-1}$ are of rank $1$. Since rank $1$ $(\varphi,\Gamma)$-moduels over $\mathcal{R}_K$ are of the form $\mathcal{R}_K(\delta)$ for some unique continuous character $\delta: \mathbb{Q}_p^{\times} \rightarrow K^{\times}$ we can describe a triangulation by a sequence of such characters $(\delta_i)_{1 \leq i \leq n}$.

So lets us say that $M$ is trianguline, is there a unique triangulation? What confuses me is that in such a case the generalized Hodge-Tate weights of $V$ correspond to the weights of the $\delta_i$ defined as $$ w(\delta_i):= \log_p \delta_i(u) / \log_p u $$ for some $u \in 1 +p \mathbb{Z}_p$ doesn't this make the triangulation unique? Where is my mistake here...

Answer 1

No, triangulations are not in general unique.

A simple way of seeing this is to consider the case when $K = \mathbf{Q}_p$, $V$ is 2-dimensional and crystalline with distinct Hodge–Tate weights, say $\{0, -r\}$ and the eigenvalues of $\varphi$ on $D_{cris}(V)$ are distinct, say $\alpha$ and $\beta$. Weak admissibility forces $v_p(\alpha)$ and $v_p(\beta)$ to be in the interval $[0, r]$ and sum to $r$. I'm going to suppose $\alpha \ne \beta$ and both $v_p(\alpha)$ and $v_p(\beta)$ are strictly positive (so $V$ is irreducible).

If you want to write down a triangulation of $V$, then the character $\delta_1$ has to be unramified, and $\delta_2$ has to restrict to $x \mapsto x^r$ on $\mathbf{Z}_p^\times$ (or maybe $x^{-r}$, I forget.) Moreover, $\delta_1(p)$ and $\delta_2(p)$ have to be $\alpha$ and $\beta$ in some order. However, we are free to choose which order, and this gives 2 distinct triangulations.

(This is exactly what the Galois reps of non-ordinary modular forms of prime-to-$p$ level and weight $\ge 2$ look like locally at $p$. The two triangulations correspond to the two $p$-stabilisations of $f$.)

Your question "Where is my mistake here..." is hard to answer, because I can't work out why you expect the assertion about HT weights to imply any uniqueness.