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 $B_{\mathrm{crys}}$?

everyoneshould be allowed to invent words...! :) $\endgroup$ – Mariano Suárez-Álvarez Jul 14 '11 at 7:17