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Wiles和Taylor精彩地证明了Fermat最后定理以来，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络的最令人震惊的证实，这些猜想是以朗兰兹计划的名义收集的，该计划将纯数学的所有分支联系在一起。更准确地说，朗兰兹程序推测，某些类型的算术对象(伽罗瓦表示)、分析对象(自同构形式)和几何对象(簇和循环)实际上都是相同的。连接这些不同世界的桥梁被称为L功能。它们是可以与上述三种类型的对象相联系的解析函数，如果两个不同世界中的两个对象具有相同的L函数，则它们对应。该项目的目的是在这种情况下，使用p-addy方法来证明几个高影响力的结果。具体地说，我们将使用p-进变形的方法，其中包括构造“族”、自同构形、伽罗瓦表示、代数圈或L函数，由帕迪克空间参数化；我们将特别使用由Andreatta、Boxer、Iovita和Pilloni发展的最新的高阶Coleman理论。我们期望应用于研究椭圆曲线的Birch和Swinnerton-Dyer猜想，自同构族的Eichler-Shimura关系，构造Euler系统，应用于岩泽主要猜想，以及新的模性结果。

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.