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

模形式在算术中一直扮演着非常重要的角色，自从怀尔斯和泰勒惊人地证明费马大定理以来，模形式的作用越来越重要。泰勒和怀尔斯的结果是对一个庞大的猜想网的最惊人的证实，这个猜想网是以朗兰兹纲领的名义收集起来的，它把纯数学的所有分支联系在一起。更准确地说，朗兰兹纲领推测，某些类别的算术对象（伽罗瓦表示）、分析对象（自同构形式）和几何对象（变异和循环）实际上都是相同的。连接这些不同世界的桥梁被称为l函数。它们是分析函数，可以与上述三种类型的对象相关联，如果两个对象在两个不同的世界中具有相同的l函数，则它们对应。该项目的目的是使用p-adic方法证明在这种情况下的几个高影响结果。特别地，我们将使用p进变形的方法，它涉及“族”的构造，自同构形式，伽罗瓦表示，代数循环或l函数，参数化的空间；我们将特别使用最近由Andreatta， Boxer， Iovita和Pilloni开发的Higher Coleman理论。我们期待应用于椭圆曲线的Birch和Swinnerton-Dyer猜想，自同构形式族的Eichler—Shimura关系，欧拉系统的构造，Iwasawa主猜想的应用，以及新的模块化结果的研究。

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.