**The International Center for Mathematical Sciences – Sofia (ICMS-Sofia)**

**presents**

## ICMS Seminar

06.08.2024, 14:00 Sofia time,

ICMS-Sofia, Room 403

#
**P-adic L-functions and the geometry of the Eigencurve**

**P-adic L-functions and the geometry of the Eigencurve**

##
Mladen Dimitrov,
University of Lille

**Abstract:**

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 Mordell–Weil rank of an elliptic curve is given by the order of vanishing of its L-function the central point. Iwasawa theory, in turn, seeks to relate the arithmetic over the p-adic cyclotomic extension with the behavior of a p-adic analytic L-function, and various recent works on the BSD conjecture rely crucially on such p-adic methods.

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.