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中的问题，这两个领域目前正在经历着令人印象深刻的发展，并且密切相关：岩泽理论和自同构型的p-进数族。我的项目有许多分支的可能性，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整数解；这是使用代数几何和分析的方法完成的。众所周知，对于大多数“动机”来说，一个人可以把两个对象联系在一起：L函数和塞尔默群(它推广了簇的有理点的概念)。一系列猜想表明，关于L函数的消失级和前导系数的信息可以转化为关于塞尔默群的信息。对于这些猜想，一种非常有效的方法是通过p进变形。给出一个动机，人们可以研究它在某些场扩展的塔上是如何变化的，比如由p^n次统一根产生的塔。在许多情况下，我们可以定义一个p-进L函数和一个扩张塔上的“大”塞尔默群。岩泽、格林伯格和贝诺瓦的猜想指出，“大”塞尔默群是具有Z_p系数的形式级数环上的(余)挠，其特征理想是由p-进L函数生成的。这通常意味着布洛赫-加藤猜想。我的研究计划的第一个目标是，在与Z.Liu(McGill)(McGill)合作的一个项目中，当常数项(即p-进L函数)消失时，可以利用Ribet的思想在Selmer群中构造余圈，从而可以用来限定它的大小，从而证明这两个理想之间的包含。这表明对于主要猜想，p-ady族是必要的，但到目前为止，只有当Shimura簇的p-普通轨迹不为空时，才构造族，这排除了许多情况。这将我们引向我的项目的第二个目标。与R.Brasca(Univerite Paris 7)一起，我们正在为没有普通基因座的PEL Shimura品种发展Hida理论。其思想是将泛p-可分群的乘法部分替换为它的整个滤除连通部分。这种方法主要利用了Dieudonne模的性质及其过滤。因此，我们接下来的项目是将其推广到不是PEL类型但具有到G-Zip堆栈的映射的Shimura变种，“具有额外结构的Dieudonne模”，例如正交的Shimura变种。期望在p元L函数的构造中得到应用。