Charlotte Clare-Hunt (University of Oxford)

There has been substantial progress in the construction of eigenvarieties and p-adic families of automorphic forms, and their relationship with Selmer groups and (p-adic) L-functions. In this talk I will introduce some of these constructions, starting with the interpolation of modular forms, and the concept of complete p-adic rigidity: the non-existence of nontrivial p-adic deformations. I will explain some of the techniques used to study the geometry of eigenvarieties, and how these specialise to show that certain noncuspidal ‘Saito—Kurokawa’ points are completely p-adically rigid. If time permits, I will also briefly outline how similar strategies may be used to construct p-adic families through cuspidal, nonholomorphic Saito—Kurokawa points and to produce nontrivial Selmer classes predicted by the Bloch—Kato conjecture.