Appears in collection : New Structures and Techniques in $p$-adic Geometry

The goal of these lectures is to discuss some aspects of the geometrization of the $p$-adic Langlands program (more precisely, the part of it which deals with locally analytic representations on topological $\mathbb{Q}_p$-vector spaces). The lectures by Hellmann (based on joint work with Hernandez and Schraen and with Heuer) will focus on the « Galois » side, introducing moduli stacks of (variants of) $p$-adic Galois representations and on the study of their geometry. The lectures of Le Bras (based on joint work with Anschütz, Rodriguez Camargo, Scholze) will focus on the « automorphic » side, explaining how to realize locally analytic representations as sheaves on a variant of Fargues-Scholze’s stack of $G$-bundles on the Fargue-Fontaine curve.