Main Conjectures for Families of Automorphic Forms

自守形式族的主要猜想

• 批准号: RGPIN-2018-04392
• 负责人: Rosso, Giovanni
• 金额: $ 1.68万
• 依托单位: Concordia University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 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=759113
• 关键词: Main; Conjectures; Families; Automorphic; Forms

My research project aims at solving problems in two areas of number theory which are currently experiencing an impressive development and are closely related: Iwasawa theory and p-adic families of automorphic forms. My project has possibilities for many ramifications which would provide interesting problems for Masters' and PhD students.One of the most important goals of number theory is the study of integral solutions of certain sets of polynomial equations; this is done using methods from algebraic geometry and analysis. It has been known that to most of "motives" one can associate two objects: the L-function and the Selmer group (which generalizes the concept of rational points of a variety). A series of conjectures state that information on the vanishing order and the leading coefficient of the L-function can be translated to information about the Selmer group. A very fruitful approach to these conjectures is via p-adic deformations. Given a motive, one can study how it varies over certain towers of field extensions, such as the tower generated by the p^n-th roots of unity. In many cases, one can define a p-adic L-function and a "big" Selmer group over the tower of extensions. Conjectures by Iwasawa, Greenberg, and Benois state that the "big" Selmer group is (co)-torsion over the ring of formal series with Z_p coefficients, and that its characteristic ideal is generated by the p-adic L-function. This often implies the Bloch--Kato conjecture. The first objective of my research program, in a joint project with Z. Liu (McGill whenever the constant term (i.e. the p-adic L-function) vanishes, one can use ideas of Ribet to construct co-cycles in the Selmer group which can be used to bound its size, and hence proving an inclusion between the two ideals. This shows that p-adic families are necessary for Main Conjectures but so far families have been constructed only when the p-ordinary locus of the Shimura variety is not empty, which excludes many cases. This leads us to the second objective of my project. With R. Brasca (Universite Paris 7) we are developing Hida theory for PEL Shimura varieties without ordinary locus. The idea is to substitute the multiplicative part of the universal p-divisible group with its whole filtered connected part. This approach uses mainly properties of the Dieudonne modules with its filtration. Hence, our subsequent project is to generalize this to Shimura varieties which are not of PEL type but with a map to the stack of G-zips, "Dieudonne modules with extra structure", such as the orthogonal Shimura varieties. Application to the construction of p-adic L-functions are expected.

我的研究项目旨在解决数论的两个领域的问题，这两个领域目前正在经历令人印象深刻的发展，并且密切相关：岩泽理论和自守形式的p进族。我的项目有可能产生许多分支，这将为硕士和博士生提供有趣的问题。数论最重要的目标之一是研究某些多项式方程组的积分解;这是使用代数几何和分析方法完成的。众所周知，对于大多数“动机”，人们可以将两个对象联系起来：L-函数和塞尔默群（它推广了簇的有理点的概念）。一系列的理论表明，L-函数的消失阶和前导系数的信息可以转化为关于塞尔默群的信息。一个非常富有成效的方法，这些结构是通过p-adic变形。给定一个动机，人们可以研究它如何在某些场扩张的塔上变化，例如由单位根的p^n次生成的塔。在许多情况下，我们可以定义一个p-adic L-函数和一个在扩张塔上的“大”塞尔默群。Iwasawa，Greenberg和Benois的猜想指出，“大”塞尔默群是Z_p系数形式级数环上的（余）-挠群，其特征理想由p-adic L-函数生成。这通常意味着Bloch-Kato猜想。我的研究计划的第一个目标，在一个与Z。Liu（麦吉尔）在常数项（即p-adic L-函数）为零时，可以利用Ribet的思想在塞尔默群中构造余圈，并利用余圈来限制其大小，从而证明两个理想之间的包含。这表明p进族对于主猜想是必要的，但到目前为止，族仅在Shimura簇的p-普通轨迹不为空时才被构造，这排除了许多情况。这就引出了我的项目的第二个目标。与R. Brasca（Universite巴黎7），我们正在发展希达理论的PEL志村品种没有普通座位。其思想是用泛p-可除群的整个滤子连通部分来代替泛p-可除群的乘法部分。该方法主要利用Dieudonne模及其过滤的性质。因此，我们随后的项目是将其推广到Shimura品种，这些品种不是PEL类型，但具有到G-zips堆栈的映射，“具有额外结构的Dieudonne模块”，例如正交Shimura品种。应用于p-adic L-函数的构造。