Abstract: We compute the Fontaine operator in the setting of Hilbert modular varieties, after taking b-cohomology, extending the work of Lue Pan in the modular curve case. As an application, we prove under some condition on weights that for overconvergent Hilbert modular forms, the partial de Rhamness condition on the Galois representation will imply the overconvergent form extends in one direction, i.e. it is "partially classical". Moreover, under generic condition, the partial classical overconvergent Hilbert modular forms vary in family.

Abstract: Let p be a prime number. Let L be a finite extension of Qp, and let E be a suficiently large finite extension of L. Let $\rho_p$ be a $2$-dimensional $E$-linear continuous representation of Gal(\bar L/L), which is de Rham with regular Hodge-Tate weights. When $\rho_p$ is of global origin, we give a strong evidence on Breuil's locally analytic $\Ext^1$ conjecture for $\rho_p$. The proof is based on a detailed geometric study of the locally analytic sections on certain completed unitary Shimura curves, \`a la Lue Pan, and a geometric realization of the translation functors on locally analytic representations.

Abstract: Recently, Yiwen Ding proposed to study the p-adic local Langlands correspondence using translation functors for locally analytic representations, which change Hodge-Tate-Sen weights on the Galois side. In this talk, we consider translations from the point of view of the categorical p-adic local Langlands correspondence. I will introduce certain maps between loci of the stack of (phi, Gamma)-modules over the Robba ring with different integral Hodge-Tate-Sen weights. We show that in the GL2(Qp) case, these maps can realize translations of (phi, Gamma)-modules geometrically.

Abstract: In this talk, we will modify the Breuil-Hellmann-Schraen's (more generally, resp., Breuil-Ding's) local model for the trianguline variety (resp.,  Bernstein paraboline variety) to certain semistable (resp., potentially semistable) non-crystalline point that have regular Hodge-Tate weights. Then we prove the existence of expected companion points on the (definite) eigenvariety and locally analytic socle conjecture for such semistable non-crystalline Galois representations, under certain hypothesis on trianguline variety and the usual Taylor-Wiles assumptions.