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，岩泽关于权重 2 的非普通模形式的理论已经取得了大量成果，这使得该理论与公认的普通情况更加接近。 然而，在权重大于 2 的情况下，在非普通情况下，人们对 Selmer 群和 L 值的最基本问题知之甚少。沿着这些路线前进的最严重障碍是（1）非-的疯狂行为普通 p 进 L 函数，以及 (2) p 处出现的局部条件的复杂性。 为了克服第一个障碍，波拉克试图对分圆 Zp 扩展的每个有限层上的非普通模形式的 Iwasawa 理论进行系统研究。 为了解决局部障碍，他的目标是使用p进局部朗兰兹对应来控制出现的局部结构，然后制定θ元素的代数理论，该理论类似于分析定义的Mazur-Tate元素。 Pollack 旨在证明一系列定理，证明这些代数定义的元素控制 Selmer 群的大小和结构，并证明主要猜想等价于这些元素与 Mazur-Tate 元素的等式。 该程序的一个特定目标是结合 Kato 的结果和代数 theta 元素理论来证明 Mazur 和 Tate 的猜想，该猜想断言他们的解析 theta 元素位于某个对偶 Selmer 群的拟合理想中。该项目的研究来自椭圆曲线理论，椭圆曲线是某些数学对象，其点具有甜甜圈的形状。 椭圆曲线曾经是纯数学家研究的焦点，现在已经在密码学的理论和实践中变得无处不在。 此外，由于安德鲁·威尔斯 (Andrew Wiles) 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。 怀尔斯定理本质上表明，人们可以在有理数（几何对象）上的椭圆曲线和权重 2 的某些模形式（微积分类型对象）之间建立一个精确的字典。 该项目超越了重量为 2 的模形式的情况，并试图对任意重量模形式的某些属性进行系统研究。 从密码学的角度来看，这些更高权重的模块化形式（可以被认为是广义椭圆曲线）是否也非常重要，还有待观察。