Geometric Aspects of the p-adic Locally Analytic Langlands Correspondence I

Explore the geometric foundations of the $p$-adic Langlands program through this advanced mathematical lecture focusing on locally analytic representations on topological $\mathbb{Q}_p$-vector spaces. Delve into the Galois side of the correspondence as you examine moduli stacks of $p$-adic Galois representations and their geometric properties, drawing from collaborative research with Hernandez, Schraen, and Heuer. Learn how this work contributes to the broader geometrization project of the $p$-adic Langlands correspondence, which seeks to understand the deep connections between Galois representations and automorphic forms through geometric methods. Discover the theoretical framework that underpins this area of arithmetic geometry, including the construction and analysis of moduli spaces that parametrize various types of $p$-adic Galois representations. This lecture serves as the first part of a series that will ultimately connect the Galois-theoretic aspects covered here with the automorphic side involving sheaves on variants of the Fargues-Scholze stack of $G$-bundles on the Fargues-Fontaine curve.