Title: Geometry of eigenvarieties and Selmer groups

Speaker: Joel Bellaiche, Columbia University

Abstract: The eigenvarieties are rigid analytic spaces that parametrizes the $p$-adic eigenforms for a given reductive group, of a given level but with various (natural or $p$-adic) weights. Their geometry, even locally, is still very poorly understand. I will explain in this talk a relation between the geometry of the eigenvarities for unitary groups at a point corresponding to a classical, non tempered, endoscopic automorphic forms on the one hand, and a Selmer group in the other. This relation may be used both way : to prove the smoothness of the eigenvariety in some cases, and to prove part of the Bloch-Kato conjecture.