Breuil–Bergdall conjecture on locally analytic vectors for crystalline trianguline representations

Let $\mathscr{S}_*^{\mathrm{cris}}$ denote the crystalline parameter space used in the paper. For $s\in\mathscr{S}_*^{\mathrm{cris}}$ non-exceptional, let $\Sigma(\delta_1,\delta_2)\mathbin{\widetilde{\oplus}}\Sigma(x^{w(s)}\delta_2,x^{-w(s)}\delta_1)$ be the amalgamated sum appearing in the natural morphism to $\Pi(s)_{\mathrm{an}}$, formed over the intertwining of the locally algebraic subrepresentations. Breuil–Bergdall conjecture. The morphism

$$\Sigma(\delta_1,\delta_2)\mathbin{\widetilde{\oplus}}\Sigma(x^{w(s)}\delta_2,x^{-w(s)}\delta_1)\longrightarrow\Pi(s)_{\mathrm{an}}$$

is a topological isomorphism. This gives an explicit description of the locally analytic vectors in the crystalline non-exceptional case. The parser marks the conjecture as resolved: the generic case was already proved by Emerton, while the cited result establishes the remaining assertion.