p-adic local Langlands and Iwasawa theory

在过去的十年里，岩泽理论的非普通模形式的重量2已经看到了激增的结果，使理论更等同于公认的普通情况。 然而，在权重大于2的情况下，在非普通的情况下，人们对塞尔默群和L-值的最基本的问题知之甚少，沿着这些路线前进的最严重的障碍是（1）非普通p进L-函数的狂野行为，以及（2）在p处出现的局部条件的复杂性质。 为了避开第一个障碍，波拉克试图在分圆Zp-扩张的每个有限水平上对岩泽非普通模形式理论进行系统的研究。 为了处理当地的障碍，他的目的是使用的p进当地朗兰兹对应控制当地的结构出现，然后制定一个代数理论的θ-元素这是类似的分析定义的马祖尔-泰特元素。 波拉克的目的是证明一系列定理表明，这些代数定义的元素控制的大小和结构的塞尔默团体，并表明，主要猜想是等同于平等的这些元素与马祖尔泰特元素。 一个特别的目标，这个计划是联合收割机的结果加藤和理论的代数θ-元素，以证明猜想马祖尔和泰特断言，他们的分析θ-元素在于拟合理想的某一双塞尔默组。动机研究这个项目来自理论的椭圆曲线是某些数学对象的点有形状的甜甜圈。 椭圆曲线曾经是纯数学家研究的焦点，现在在密码学的理论和实践中都无处不在。 此外，由于安德鲁·怀尔斯在90年代中期对费马大定理的突破性证明，我们现在知道椭圆曲线与模形式密切相关，模形式是具有许多对称性的复变量的函数。 怀尔斯定理本质上指出，人们可以在有理数上的椭圆曲线（几何对象）和某些重量为2的模形式（微积分类型的对象）之间建立一个精确的字典。 这个项目超越了重量2模形式的情况，并试图对任意重量模形式的某些性质进行系统的研究。 从密码学的角度来看，这些更高权重的模形式（可以被认为是广义椭圆曲线）是否也非常重要还有待观察。

Over the last decade, the Iwasawa theory of non-ordinary modular forms of weight 2 has seen a surge of results which has brought the theory more on par with the well-established ordinary case. However, in weights greater than 2, little is known in the non-ordinary case about even the most fundamental questions on Selmer groups and L-values.The most serious obstacles to progress along these lines are (1) the wild behavior of non-ordinary p-adic L-functions, and (2) the complicated nature of the local conditions at p which arise. To circumvent the first of these obstacles, Pollack seeks to make a systematic study of the Iwasawa theory of non-ordinary modular forms at each finite level of the cyclotomic Zp-extension. To deal with the local obstacle, he aims to use the p-adic local Langlands correspondence to control the local structures which arise, and to then formulate an algebraic theory of theta-elements which is the analogue of the analytically defined Mazur-Tate elements. Pollack aims to prove a series of theorems which show that these algebraically defined elements control the size and structure of Selmer groups, and to show that the main conjecture is equivalent to the equality of these elements with Mazur-Tate elements. One particular goal of this program is to combine results of Kato and the theory of algebraic theta-elements to prove a conjecture of Mazur and Tate which asserts that their analytic theta-element lies in the Fitting ideal of a certain dual Selmer group.The motivation for the study of this project comes from the theory of elliptic curves which are certain mathematical objects whose points have the shape of a doughnut. Elliptic curves, once the focus of study of only pure mathematicians, have now become ubiquitous in both the theory and practice of cryptography. Further, as a result of the breakthrough proof of Fermat's Last Theorem by Andrew Wiles in the mid 90s, we now know that elliptic curves are intimately connected to modular forms which are functions of a complex variable with many many symmetries. Wiles' theorem essentially states that one can make a precise dictionary between elliptic curves over the rational numbers (which are geometric objects) and certain modular forms of weight 2 (which are calculus-type objects). This project pushes out beyond the case of weight 2 modular forms, and seeks to make a systematic study of certain properties of arbitrary weight modular forms. It remains to be seen if these higher weight modular forms (which can be thought of as generalized elliptic curves) are also highly important from a cryptographic viewpoint.