Intro : In contemporary arithmetic geometry, the interplay between p-adic L-functions and p-adic Hodge theory is of paramount importance. While p-adic L-functions encode deep arithmetic information about special values of L-functions (Iwasawa Main Conjecture and BSD Conjecture), Fontaine's p-adic Hodge theory provides the essential "linear algebra" data to classify p-adic Galois representations. Connecting these two fields allows for the construction of L-functions in settings where classical methods fall short, particularly for non-ordinary forms and general families of motives.

Subject : This study group is structured to bridge these classical approaches with the modern cohomological framework, primarily following Colmez's course notes(we take it as base and provide more content). We will begin by reviewing the classical construction of the Kubota-Leopoldt p-adic zeta function and modular forms, utilizing modular symbols. Subsequently, we will introduce the fundamental objects of p-adic Hodge theory—Fontaine's period rings and (φ,Γ)-modules. The core objective is to reinterpret and generalize p-adic L-functions using these tools. We will cover Tate duality, the explicit reciprocity laws, and finally, use Sen’s theory and overconvergence to understand the p-adic interpolation of algebraic data for broader families of motives. 

We will focus on p-adic L-function. This means we treat p-adic Hodge theory from a view of p-adic L-function.

When : Saturday, 15h00-18h00, Sophie Germain, room 633 

Syllabus (update after every talk) : PDF, link of overleaf.