p-adic variation in Iwasawa theory

数论在过去几十年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、伽罗瓦表示法还是l值）放入p进族中，他就能更好地理解这个对象。模块化形式适合Hida/Coleman家庭；伽罗瓦表示由通用变形空间参数化；l值由p进l函数插值。在这个提议中，波拉克试图研究p进变量的三个不同实例。首先，他的目的是给一个猜想公式的切环模不变量的倒钩特征形式。他的方法是将信息从爱森斯坦级数的p进族传递到尖形的希达族，以计算相关的mu不变量。其次，他的目标是研究加藤？s欧拉系统附加到一个模形式。他的方法的一个新颖方面是在伽罗瓦变形空间上改变欧拉系统。在这个空间中，普通形式和非普通形式交织在一起，他希望将已知的岩川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进变量相关的计算项目，包括Hida族和权重1形式的计算。数论是最古老的数学领域之一，它经常借用其他数学领域的方法来解决与数字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，尽管它具有离散的性质，但它对数论产生了深远的影响。本提案探讨了试图更好地理解某些数论对象的想法，即模形式，通过将它们放入一个家族并将其作为一个整体进行研究。想象一下，研究蚂蚁的行为和研究蚂蚁在蚁群中的角色有什么不同。数论中族的概念是通过p进分析得到的。p进分析是模算法的一种推广，模算法本身因其在公钥密码学中的作用而获得了极大的公众声誉，公钥密码学是互联网商业的支柱。因此，通过研究模形式族，本项目提出的工作目标是在理论和计算层面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加深都可能有潜在的密码学应用。

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.