In this final lecture I will explain that in high degree the Abel-Jacobi morphism in my context is a pro-étale locally trivial fibration in simply connected diamonds. As a corollary I deduce my geometrization conjecture for GL_1. I will explain how this is linked to the classical local class field theory.

Laurent Fargues is Directeur de Recherche at CNRS / Institut de mathe ́matiques de Jussieu. He is a leading expert in questions on the local Langlands correspondence, Rapoport-Zink spaces, and p-adic Hodge theory, and has made striking contributions to all of these areas. His work is characterized by a unique vision of how these subjects are related, and his insights are shaping these fields. He received the "Petit d’Ormoy, Carrière, Thébault", prize given by the French Academy of Science. He is an invited sessional speaker at the upcoming 2018 ICM at Rio de Janeiro, Brazil.