For centuries, understanding special values of L-functions has been a significant research topic in number theory. Their study has been central to many celebrated pieces of mathematics, from Dirichlet's theorem on primes in arithmetic progressions and the class number formula to the Riemann hypothesis and the Birch and Swinnerton-Dyer (BSD) conjecture, two of the famous millennium problems.
The BSD conjecture predicts that the rank of points on an elliptic curve over a number field should be given by the order of vanishing of its L-function at s = 1. More recently, Bloch and Kato conjectured a relationship between the analytic properties of L-functions of motives (or cuspforms over a reductive group via the Langlands program) and the arithmetic of the attached p-adic Galois representation. Iwasawa theory, in turn, seeks to relate the arithmetic over the p-adic cyclotomic extension, and the behavior of the p-adic analytic L-function. While trivial zeros of p-adic L-functions and their L-invariants were considered by Mazur, Tate and Teitelbaum in their quest to formulate a p-adic analogue of the BSD conjecture, various recent works on the Bloch–Kato conjecture rely crucially on p-adic L-functions and the Iwasawa Main Conjecture.
An amazing feature of the p-adic L-functions is their ability to live in families, thus their laws are governed by the geometry of p-adic eigenvarieties. In this lecture we will illustrate this philosophy through examples coming from classical modular forms and the Coleman-Mazur eigencurve.