p-adic methods in number theory: eigenvarieties and cohomology of Shimura varieties for the study of L-functions and Galois representations

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Modular forms have always been playing a very important role in arithmetic, and since the spectacular proof of Fermat Last Theorem by Wiles and Taylor, their role is more and more fundamental. The results of Tylor and Wiles are the most astonishing confirmation of a huge web of conjectures, collected under the name of Langlands program, which ties all branches of pure mathematics. More precisely, the Langlands program conjectures that certain classes of arithmetic objects (Galois representations), of analytic objects (automorphic forms), and of geometric objects (varieties and cycles) are in reality all the same. The bridges connecting these different worlds are called L-functions. They are analytic functions that can be associated with the three aforementioned types of objects, and two objects in two different worlds correspond if they have the same L-function. The aim of the project is to prove several high-impact results in this setting using p-adic methods. In particular, we will use methods of p-adic deformations, which involve the construction of "families", of automorphic forms, of Galois representation, of algebraic cycles or of L-functions, parameterized by padic spaces; we will use in particular the very recent Higher Coleman Theory developed by Andreatta, Boxer, Iovita, and Pilloni. We expect applications to the study of the conjectures of Birch and Swinnerton-Dyer for elliptic curves, Eichler--Shimura relations for families of automorphic forms, construction of Euler system, and application to Iwasawa Main Conjecture, and new modularity results.