The socle conjecture for locally analytic representations

Let $v\in S_p$. For each $v$, let $\widetilde v$ be the chosen place above $v$, let $\rho_{\widetilde v}$ be the associated generic potentially crystalline Galois representation, let $\mathscr F_{\widetilde v}$ be a minimal parabolic filtration of $\texttt r(\rho_{\widetilde v})$, and let $C(w_{\widetilde v},\mathscr F_{\widetilde v})$ be the associated locally analytic representation. Write $\mathscr W^{P_{\widetilde v}}_{\min,F_{\widetilde v}}$ for the corresponding set of minimal Weyl-group representatives and $w_{\mathscr F_{\widetilde v}}$ for the element attached to the filtration. Socle conjecture. For $w_{\widetilde v}\in \mathscr W^{P_{\widetilde v}}_{\min,F_{\widetilde v}}$, the representation

$$\widehat{\otimes}_{v\in S_p}C(w_{\widetilde v},\mathscr F_{\widetilde v})$$

is a subrepresentation of $\widehat S(U^p,E)[\mathfrak m_\rho]$ if and only if

$$w_{\widetilde v}\leq w_{\mathscr F_{\widetilde v}}w_{0,F_{\widetilde v}}\quad\text{for all }v\in S_p.$$

This is the socle conjecture, a special case of a conjecture of Breuil. It is known in the crystalline case under a Taylor–Wiles hypothesis, while the non-trianguline case has only partial results in the source and is the subject of the paper.