Consider a finite-dimensional $\mathbf{Q}\_p$-vector space $V$ and a continuous representation $\rho : G\_{\mathbf{Q}\_p} \to \mathrm{GL}(V)$. Fontaine introduced various $\mathbf{Q}\_p$-algebras with $G\_{\mathbf{Q}\_p}$-actions, notated $B\_{\bullet}$ where $\bullet \in \left\{\mathrm{crys}, \mathrm{st}, \mathrm{dR}\dots \right\}$, which "classify" interesting representations $V$; we say $V$ is $\bullet$ if equality holds in the relation $\mathrm{dim}(B\_{\bullet} \otimes V)^{G\_{\mathbf{Q}\_p}} \leq \dim{V}$.


Recently, the notion of a "trianguline" Galois representation has become increasingly important. Avoiding the precise definition, I will say rather perversely that the trianguline representations are roughly the closure of the crystalline locus in the set of all $\rho$'s as above (made precise, this is a theorem of Chenevier which is of course predicated on the actual definition of trianguline). So my question: *is there a ring of periods $B\_{\mathrm{tri}}$ which classifies trianguline representations in the above manner, and/or is such a ring expected to exist?* Is/should it be a suitable "completion" of $B\_{\mathrm{crys}}$?



Is there a "trianguline period ring", or is one expected?
The category of trianguline representation is stable under all the usual representation-theoretic operations (subs, quotients, $\oplus$, $\otimes$), so by some general tannakian formalism, there does exist a ring $B\_{tri}$. The rough idea is to look at $Q\_p^{alg} \otimes B\_{st} \langle \langle \log(t) \rangle \rangle$ where "$\langle \langle \log(t) \rangle \rangle$" means "power series with some non zero radius of convergence" and $t$ is the usual $t$ in this business. It's interesting to note that I first heard about this ring from Fontaine (around 2003-04 maybe - I was still at Harvard) when trianguline representations had not yet been defined. Fontaine told me at the time that repns admissible for this ring should be interesting! A few comments are in order: 


* $B\_{st}$ does not have the structure of a Banach space, so you need to figure out what "radius of convergence" means
* $\exp(\log(t))=t$, so there are relations in the definition of your ring
* you need to decide if you want a ring for "trianguline" or "split-trianguline" repns


I thought about this again a few weeks ago and, if I remember correctly, came to the conclusion that if you take $B = Q\_p(\mu\_p) \otimes \hat{Q}\_p^{nr} \otimes B\_e \otimes Q\_p \langle \langle \log(t) \rangle \rangle \otimes Q\_p[\log(\tilde{p})]$ (whew!), where $B\_e=B\_{cris}^{\phi=1}$, then $B$-adm reps of $G\_{Q\_p}$ are trianguline, and conversely split trianguline reps of $G\_{Q\_p}$ with integer slopes are $B$-adm. This hopefully gives an idea of the kind of ring which one should be looking for.


