Wednesday, September 15, 2021 12:30 AM

Coleman and Mazur constructed the first eigencurve, a $p$-adic analytic space which parametrizes $p$-adic overconvergent modular eigenforms of nonzero eigenvalue under the Hecke operator $U_p$. Since then, many authors have constructed analogous higher-dimensional eigenvarieties for various other kinds of automorphic forms. We will talk about $p$-adic modular forms, their generalization to Hilbert modular forms, and geometric questions about the resulting eigenvarieties.