p-adic local Langlands and Iwasawa theory

p-进局部 Langlands 和 Iwasawa 理论

• 批准号: 1001768
• 负责人: Robert Pollack
• 金额: $ 20.04万
• 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: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001768&HistoricalAwards=false
• 关键词: adic; local; Langlands; Iwasawa; theory

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.

在过去的十年中，非常见的重量模块化形式的iwasawa理论看到了一系列的结果，这使该理论更与公认的普通案例相提并论。 然而，在非平凡的情况下，关于Selmer组和L值最基本的问题，在重量大于2的重量中，几乎没有人知道。沿着这些线路进行进展的最严重的障碍是（1）非常见的P-ADIC L-functions的狂野行为，以及（2）p的本地条件的复杂性质。 为了避免这些障碍中的第一个，Pollack试图在环环ZP扩展的每个有限水平上对硫磺的非常见模块化形式理论进行系统的研究。 为了应对当地的障碍，他的目标是使用P-AdiC的局部兰兰人来控制出现的局部结构，然后制定一个代数的theta元素理论，这是分析定义的Mazur-Tate元素的类似物。 Pollack旨在证明一系列定理表明这些代数定义的元素控制Selmer组的大小和结构，并表明主要猜想等于与Mazur-Tate元素的这些元素的平等性。 该计划的一个特殊目标是结合KATO的结果和代数theta元素的理论，证明了Mazur和Tate的猜想，这断言它们的分析性基于元素在于某个二元selmer群体的拟合理想。该项目的动机来自该项目的动机，这是椭圆形的数学对象的一个​​贡献，它是一个贡献的一定的贡献。 椭圆形曲线曾经是对纯数学家的研究的重点，现在在密码学的理论和实践中都变得无处不在。 此外，由于安德鲁·威尔斯（Andrew Wiles）在90年代中期的《安德鲁·威尔斯》（Andrew Wiles）的最后一个定理的突破性证明，我们现在知道椭圆曲线与模块化形式密切相关，这是模块化形式，这是与许多对称性的复杂变量的函数。 威尔斯定理基本上指出，一个人可以在椭圆形曲线（几何对象）和某些重量2的模块化形式（这是colculus-type对象）之间制作精确的词典。 该项目推出了重量2模块化形式的情况，并试图对任意权重模块化形式的某些特性进行系统的研究。 从加密角度来看，这些更高的重量模块化形式是否也非常重要，还有待观察。