MoGaDAuRe

This project is at the heart of the Langlands program, at the interface between Galois and automorphic representations. The goal is to study families of both such representations using methods of geometric representation theory. Precisely, we will first study the difference between these two objects from a geometric point of view by studying the density of automorphic points in deformation spaces of global Galois representations. Then, we will give a new construction of Eigenvarieties, i.e. families of automorphic representations, using rigid analytic D-modules, generalizing previous construction and introducing new families with specific arithmetic properties. Finally, we will study a local version of moduli spaces of Galois representations : the stacks of $(\Phi,\Gamma)$-modules, and its (derived) category of coherent sheaves. We hope to construct a part of the (emerging) Categorical Langlands correspondance, with values in this category. These three objectives are interrelated : for example using families we can study the density of automorphic points via the socalled infinite fern, and families will give rise to global counterparts of derived coherent sheaves on the stack of $(\Phi,\Gamma)$-modules, whose geometry can be used to give back informations on the geometry of Eigenvarieties.