This is a follow up of the conference organized last year around similar theme. As in the previous conference, we are interested in the $p$-adic properties of modular forms.
The general framework for the study of congruences between modular forms is provided by the theory of $p$-adic modular forms developed in fundamental papers of Serre, Katz, Hida, Ribet, Mazur and Coleman (1970's-1990's).
Given a modular form, we associate a Galois representation to that by a classical theorem due to Deligne. We can restrict this Galois representation to different decomposition groups. There is a recent interest to understand the representation for the case $l=p$. Starting with the work of Breuil to formulate mod $p$ and $p$ adic Langlands programme similar to the usual Langlands programme. This representations appear in the cohomology groups of modular curves. Recent development on the subject will be studied for different groups.
In the other direction, we will be interested in understanding the Eisenstein part of this cohomology groups of the modular curves and the applications in the context of $p$-adic automorphic forms. Various applications of the theme including Ogg's conjecture will be discussed in details. Young researchers and early career mathematicians are encouraged to apply.
List of mini-courses:
1. Fred Diamond (King's college, London), Serre's conjecture for totally real fields.
2. Benjamin Schrean (Orsay, Paris), Mod p Local Langlands for Gl_2/F for totally real fields.
3. Stefano Morra (University of Paris 8) Completed cohomology for Gl_2/F for totally real fields.
Eligibility criteria : Graduate students in algebraic number theory