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...



Trianguline representation
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.


