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.

合作奖的摘要DMS-0439264和Pollack和Weston的DMS-0440708：Iwasawa理论涉及算术对象的Galois理论和P-ADIC分析方面之间的关系。 研究人员的植物可以研究具有超高还原的模块化形式理论。 这项工作的主要重点是Iwasawainvariants在模块化形式的P-Adic分析家族中的行为，例如Coleman和Mazur的Theigencurve。 实际上，在普通情况下众所周知，这些变化的定义在一般情况下尚不清楚。 为了定义和研究代数瓦沙瓦不变，PI的打算使用P-Adic Hodge理论Offontaine来研究模块化形式的过度环球球场的Selmer群体的生长。 分析不变的定义应与模块化L功能的特殊值息息相关；在家庭中表现出所需的行为将涉及对模块化L功能的特殊价值观的一致性研究，这是最近许多工作中反复出现的主题。 该项目的无目的目标是表明，可以通过检查家庭中的Asingle形式来检查整个家庭的主要猜想。 研究人员还打算研究Kida，Hachimori andMatsuno的Riemann-Hurwitz类型公式的一般公式，描述了P- Adic iwasawa不变性数量的变化数量领域的过度延伸，对较高的重量理论的模态形式，通常考虑到了较高的数学训练，而较早的数学是对纪念性的良好的分类时期的，这是一个不分解的时期。这些应用涉及称为椭圆形的算术几何对象。 模块化形式是该提案中研究的主要对象，是椭圆曲线的一般化，在现代数字理论中起着基本作用。 这项提案协议中调查的问题与密码学关键有关的问题可能会产生对它们的见解。