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 $B_{\mathrm{crys}}$?
Best Answer
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:
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.