Locally analytic ${\mathrm{E}\mathrm{x}\mathrm{t}}^1$ for ${\mathrm{G}\mathrm{L}}_2({\mathbb{Q}}_p)$ in de Rham non trianguline case

We prove Breuil’s conjecture on locally analytic ${\mathrm{E}\mathrm{x}\mathrm{t}}^1$ for ${\mathrm{G}\mathrm{L}}_2({\mathbb{Q}}_p)$ in de Rham non-trianguline case.