p-adic variation in Iwasawa theory

岩泽理论中的 p 进变分

• 批准号: 1303302
• 负责人: Robert Pollack
• 金额: $ 16.3万
• 依托单位: Trustees of Boston University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1303302&HistoricalAwards=false
• 关键词: adic; variation; Iwasawa; theory

A major theme in number theory which has emerged over the last several decades is that one can better understand a given arithmetic object (be it a modular form, Galois representation, L-value) if one can put this object into a p-adic family. Modular forms fit into Hida/Coleman families; Galois representations are parametrized by universal deformation spaces; L-values are interpolated by p-adic L-functions. In this proposal, Pollack seeks to study three different instances of p-adic variation. First, he aims to give a conjectural formula for the cyclotomic mu-invariant of a cuspidal eigenform. His method is to transfer information from a p-adic family of Eisenstein series to the Hida family of the cuspform in order to calculate the relevant mu-invariant. Second, he aims to study the variation in p-adic families of Kato?s Euler system attached to a modular form. A novel aspect to his approach is to vary the Euler system over a Galois deformation space. As ordinary and non-ordinary forms are intertwined in this space, he hopes to transfer known Iwasawa theoretic information from the ordinary case to the non-ordinary case. Third, he proposes a number of computational projects relating to p-adic variation including the computation of Hida families and of weight one forms.Number theory, one of the oldest fields of mathematics, has often borrowed methods from other mathematical fields to attack questions relating to the basic properties and patterns of numbers. For instance, calculus is the study of functions of a continuous variable, but nonetheless has had a profound impact on number theory despite its very discrete nature. This proposal explores the idea of trying to better understand certain number theory objects, namely modular forms, by putting them into a family and studying the family as a whole. Imagine the difference between studying the actions of an ant versus the role of an ant in its colony. The notion of a family in number theory is done through p-adic analysis. P-adic analysis is a generalization of modular arithmetic which itself gained great public fame in its role in public key cryptography, the backbone of internet commerce. Thus, by studying families of modular forms, the goal of the work proposed in this project is to deepen our understanding of any individual modular form on both a theoretical and computational level. Modular forms are intimately related to elliptic curves which are also very useful in cryptography and are commonly used to encrypt cellular transmissions. Thus any deepening of our understanding of modular forms could have potential cryptographic applications.

在过去的几十年中出现的一个主要主题是，如果可以将此对象放入P-Adic家族中，那么人们可以更好地理解给定的算术对象（无论是模块化的形式，Galois表示，L值）。模块化形式适合HIDA/Coleman家庭； GALOIS表示由通用变形空间参数化； l值通过p-adic l功能插值。 在此提案中，Pollack试图研究三种不同的P-ADIC变异实例。首先，他的目标是为cuspidal特征形式的环流MU不变提供一个猜想的公式。 他的方法是将信息从艾森斯坦系列的P-Adic家族转移到Cuspform的HIDA家族，以计算相关的MU-Invariant。 其次，他的目标是研究与模块化形式相关的加藤欧拉系统的P-Adic家族的变化。 他的方法的一个新颖方面是在galois变形空间上改变欧拉系统。 由于普通和非平凡的形式在这个空间中交织在一起，他希望将已知的伊瓦沙理论信息从普通情况转移到非常见情况。 第三，他提出了许多与P-ADIC变异有关的计算项目，包括计算HIDA家族和权重的形式。数学最古老的数学领域之一，经常从其他数学领域借用了与基本数字的基本属性和数字有关的攻击问题。 例如，微积分是对连续变量功能的研究，但是尽管具有非常离散的性质，但仍对数字理论产生了深远的影响。 该建议探讨了试图更好地理解某些数字理论对象的想法，即模块化形式，通过将它们放入家庭中并研究整个家庭。 想象一下，研究蚂蚁的作用与蚂蚁在其菌落中的作用之间的区别。 家庭数理论的概念是通过P-ADIC分析来完成的。 P-ADIC分析是模块化算术的概括，它本身在公共密钥密码学（Internet Commerce的骨干）中获得了很大的公开声誉。 因此，通过研究模块化形式的家庭，该项目提出的工作的目标是加深我们对理论和计算水平上任何单个模块化形式的理解。 模块化形式与椭圆曲线密切相关，椭圆曲线也非常有用，在密码学中也非常有用，通常用于加密细胞传播。 因此，对我们对模块化形式的理解的任何加深都可以具有潜在的加密应用。

项目成果

Slopes of Modular Forms and the Ghost Conjecture

模形式的斜率和幽灵猜想

• DOI: 10.1093/imrn/rnx141
• 发表时间: 2017
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Bergdall, John;Pollack, Robert
• 通讯作者: Pollack, Robert

A remark on non-integral p-adic slopes for modular forms

关于模形式的非积分 p-adic 斜率的评论

• DOI: 10.1016/j.crma.2017.01.012
• 发表时间: 2017
• 期刊: Comptes Rendus Mathematique
• 影响因子: 0.8
• 作者: Bergdall, John;Pollack, Robert
• 通讯作者: Pollack, Robert

Slopes of modular forms and the ghost conjecture, II

模形式的斜率和幽灵猜想，II

• DOI: 10.1090/tran/7549
• 发表时间: 2019
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Bergdall, John;Pollack, Robert
• 通讯作者: Pollack, Robert