Families of modular forms and applications to Iwasawa theory

Iwasawa theory seeks to formulate and prove refined versions of the analytic class number formula and the Birch–Swinnerton–Dyer formula. A key modern tool in these developments is the use of p-adic families of modular forms, which make it possible to construct and interpolate the p-adic L-functions and Euler systems underlying such refinements. In this talk, which I will aim to keep understandable for a general audience in number theory, I will outline how these families are built and why they are so effective in Iwasawa theory. If time permits, I will conclude with some recent results of my own on the geometry of spaces of p-adic modular forms.