The Kudla program exploits deep connections between algebraic cycles and derivatives of families of Eisenstein series. In the p-adic setting, a richer supply of p adic families is encoded by eigenvarieties — rigid analytic spaces that p-adically interpolate systems of Hecke eigenvalues arising from spaces of automorphic forms. In this course, we'll give a brief introduction to this theory, focusing on the original example, the eigencurve constructed by Coleman and Mazur.

• Lecture 1: Construction of eigenvarieties. We will introduce Buzzard's general formalism for constructing eigenvarieties, the so-called eigenvariety machinery. We will then specialise to the case of the Coleman–Mazur eigencurve.
• Lecture 2: First properties of the eigencurve. We will discuss key properties of the eigencurve: the density of classical points, reducedness, and the family of pseudorepresentations over the eigencurve.
• Lecture 3: Local geometry at classical points. We will discuss results on the local geometry of the eigencurve at classical points, with particular focus on the geometry at classical weight-one points as described by Bella¨ıche and Dimitrov. We will then discuss the connections to p-adic transcendence theory and the arithmetic of S-units in number fields, as well as their applications.

Prerequisites: Familiarity with rigid analytic spaces, at the level of Several Approaches to Non-Archimedean Geometry (B. Conrad).