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

数论中的 p-adic 方法：用于研究 L 函数和伽罗瓦表示的 Shimura 簇的特征簇和上同调

• 批准号: 577144-2022
• 负责人: Rosso, GiovanniG
• 金额: $ 3.28万
• 依托单位: Concordia University
• 依托单位国家: 加拿大
• 项目类别: Alliance Grants
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758841
• 关键词: adic; methods; number; theory; eigenvarieties

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.

模块化形式一直在算术中起着非常重要的作用，并且由于威尔斯和泰勒的壮观证明了Fermat的壮观证明，因此它们的作用越来越基本。 Tylor和Wiles的结果是以Langlands计划的名义收集的巨大猜想的最令人惊讶的确认，该猜想与纯数学的所有分支联系在一起。更确切地说，兰兰兹计划猜想某些类别的算术对象（GALOIS表示），分析对象（自动形式）和几何对象（品种和周期）实际上都是相同的。连接这些不同世界的桥梁称为L功能。它们是可以与上述三种对象类型相关联的分析函数，如果它们具有相同的L功能，则两个不同世界中的两个对象对应。该项目的目的是使用P-ADIC方法在这种情况下证明几个高影响力结果。特别是，我们将使用P-ADIC变形的方法，涉及构建“家族”，自动形式，Galois代表，代数周期或L功能的构建，由PADIC空间参数化；我们将特别使用Andreatta，Boxer，Iovita和Pilloni开发的最新更高的Coleman理论。我们希望将桦木和Swinnerton-Dyer猜想的研究应用于椭圆形曲线，Eichler-shimura关系，用于自其形式的家族，欧拉系统的建设以及对伊瓦沙瓦主要猜想的应用以及新的模块化结果。