On the locally analytic ${\text{Ext}}^1$-conjecture in the ${\text{GL}}_2(L)$ case

Let $L$ be a finite extension of ${\mathbb{Q}}_p$. We calculate the dimension of ${\text{Ext}}^1$-groups of certain locally analytic representations of ${\text{GL}}_2(L)$ defined using coherent cohomology of Drinfeld curves. Furthermore, let ${\rho }_p$ be a $2$-dimensional continuous representation of $\text{Gal}(\bar{L}\mathrm{/}L)$, which is de Rham with parallel Hodge-Tate weights $0,1$ and whose underlying Weil-Deligne representation is irreducible. We prove Breuil’s locally analytic ${\text{Ext}}^1$ conjecture for such ${\rho }_p$. As an application, we show that the isomorphism class of the multiplicity space ${\mathrm{\Pi }}_{\text{geo}}^{\text{an}}({\rho }_p)$ of ${\rho }_p$ in the pro-étale cohomology of Drinfeld curves uniquely determines the isomorphism class of ${\rho }_p$.