I will review the construction and the structure of the fundamental curve of p-adic Hodge theory introduced and studied in my joint work with Jean-Marc Fontaine. This curve has different flavors and can either be considered as the analog of a projective smooth algebraic curve (a Dedekind scheme over the p-adic numbers but not of finite type) or the analog of a compact Riemann surface (a quasicompact partially proper adic space over the p-adic numbers that is not topologically of finite type). In the meantime I will explain some elementary structure results for "holomorphic functions of the variable p".

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.