Collaborative Research: P-adic Variation of Supersingular Iwasawa Invariants

合作研究：超奇异Iwasawa不变量的P进变分

• 批准号: 0439264
• 负责人: Robert Pollack
• 金额: $ 11.11万
• 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: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0439264&HistoricalAwards=false
• 关键词: Collaborative; Research; adic; Variation; Supersingular

Abstract for collaborative award DMS-0439264 and DMS-0440708 of Pollack and Weston:Iwasawa theory is concerned with the relation between the Galois theoreticand p-adic analytic aspects of arithmetic objects. The investigatorspropose to study the Iwasawa theory of modular forms with supersingularreduction. A primary focus of this work is the behavior of Iwasawainvariants in p-adic analytic families of modular forms such as theeigencurve of Coleman and Mazur. In fact, the definitions of theseinvariants, well known in the ordinary case, are not yet known in generalin the supersingular case. In order to define and study the algebraicIwasawa invariants the PI's intend to use the p-adic Hodge theory ofFontaine to study the growth of Selmer groups of modular forms overcyclotomic fields. The definitions of the analytic invariants should berelated to special values of modular L-functions; exhibiting the desiredbehavior in families will involve a study of congruences of special valuesof modular L-functions, a recurring theme in much recent work. Aneventual goal of this project is to show that the main conjecture ofIwasawa theory can be checked for an entire family by checking it for asingle form in the family. The investigators also intend to study thegeneralization of the Riemann-Hurwitz type formula of Kida, Hachimori andMatsuno, which describe the change in p-adic Iwasawa invariants overp-extensions of number fields, to modular forms of higher weight.Number theory, often considered the oldest mathematical discipline, has inrecent times developed remarkable applications to cryptography. Many ofthese applications involve arithmetic geometric objects known as ellipticcurves. Modular forms, the primary object of study in this proposal, aregeneralizations of elliptic curves which play a fundamental role in modernnumber theory. The questions investigated in this proposal deal withinvariants which are closely related to those of interest in cryptographyand may perhaps yield some insight into them.

波拉克和韦斯顿的合作奖DMS-0439264和DMS-0440708摘要：岩泽理论关注的是伽罗瓦理论和算术对象的p-adic分析方面之间的关系。 他们建议研究具有超奇异约化的模形式的岩泽理论。 这项工作的一个主要重点是行为Iwasawainvariants在p-adic分析家庭的模块化形式，如theeigencurve的科尔曼和马祖尔。 事实上，这些不变量的定义，众所周知，在普通情况下，还不知道一般在超奇异的情况下。 为了定义和研究代数Iwasawa不变量的PI打算使用的p进霍奇理论的Fontaine研究增长的塞尔默集团的模块形式overcyclotomic领域。 解析不变量的定义应该与模L-函数的特殊值有关;在族中表现出期望的行为将涉及对模L-函数的特殊值的同余的研究，这是最近许多工作中反复出现的主题。 这个项目的最终目标是表明，主要猜想ofIwasawa理论可以检查整个家庭通过检查它的一个单一的形式在家庭。 研究人员还打算研究Kida，Hachimori和Matsuno的Riemann-Hurwitz型公式的推广，该公式描述了数域的p-扩展上的p-adic Iwasawa不变量到更高权重的模形式的变化。数论，通常被认为是最古老的数学学科，最近在密码学中有了显着的应用。这些应用程序中的许多涉及算术几何对象称为椭圆曲线。 模的形式，在这个建议的主要研究对象，是广义的椭圆曲线，发挥了重要作用，在现代数论。 在这个建议中调查的问题涉及不变量，这些不变量与密码学中感兴趣的不变量密切相关，并且可能会对它们产生一些见解。