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.

在过去的几十年里，数论中出现的一个主要主题是，如果一个人能把一个给定的算术对象(无论是模形式、伽罗瓦表示法、L值)放在一个p-进数族中，他就能更好地理解这个对象。模形式适用于Hida/Coleman族；伽罗瓦表示由泛形变空间参数化；L值由p元L函数插值。在这项提议中，波拉克试图研究三种不同的p-进位变异实例。首先，他的目的是给出一个关于尖锥本征形的割圆不变量的猜想公式。他的方法是将信息从Eisenstein级数的p-进数族传递到尖形的Hida族，以计算相关的u不变量。其次，他的目的是研究依附于模形式的加藤-S欧拉系统的p-进家庭的变异性。他的方法的一个新方面是在伽罗华形变空间上改变欧拉系统。由于普通和非普通形式在这个空间中交织在一起，他希望将已知的岩泽理论信息从普通情况转移到非普通情况。第三，他提出了一些与p元变分有关的计算项目，包括Hida族的计算和权一型的计算。数论是数学中最古老的领域之一，它经常借用其他数学领域的方法来解决与数的基本性质和模式有关的问题。例如，微积分是对连续变量函数的研究，但尽管具有非常离散的性质，它仍然对数论产生了深远的影响。这项建议探索了这样一种想法，即试图通过将某些数论对象放入一个家庭并将其作为一个整体进行研究来更好地理解它们，即模形式。想象一下，研究蚂蚁的行为与研究蚂蚁在蚁群中扮演的角色有什么不同。数论中的族的概念是通过p元分析来实现的。P-Add分析是模算术的推广，它本身在公钥密码学中的作用获得了很大的声誉，公钥密码学是互联网商业的支柱。因此，通过研究模型族，本项目提出的工作的目标是在理论和计算水平上加深我们对任何单个模形式的理解。模形式与椭圆曲线密切相关，椭圆曲线在密码学中也非常有用，通常用于加密蜂窝传输。因此，我们对模形式理解的任何深化都可能具有潜在的密码学应用。

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