Collaborative Research: P-adic Variation of Supersingular Iwasawa Invariants

波拉克和韦斯顿的合作奖DMS-0439264和DMS-0440708的摘要：岩泽理论涉及算术对象的伽罗瓦理论和p-进分析方面之间的关系。研究人员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焦点是模形式的p-进解析族中的Iwawav不变量的行为，例如Coleman和Mazur的特征曲线。事实上，这些不变量的定义在普通情况下是众所周知的，但在超奇异情况下还不是普遍知道的。为了定义和研究代数岩泽不变量，Pi打算利用Fontaine的p-进Hodge理论来研究模型上割圆域的Selmer群的增长性。解析不变量的定义应该与模L函数的特殊值有关；要在家庭中表现出期望的行为，将涉及到对模L函数特殊值的同余的研究，这是最近工作中反复出现的主题。这个项目的一个新的最终目标是证明岩泽理论的主要猜想可以通过检查整个家庭的单一形式来检验。研究人员还打算研究Kida、Hachimori和Matsuno的Riemann-Hurwitz型公式的推广，这些公式描述了从数域的扩展上的p进Iwaawa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，近年来在密码学中得到了显著的应用。其中许多应用涉及称为椭圆曲线的算术几何对象。模形式是这一建议的主要研究对象，它是椭圆曲线的重新生成，它在现代数论中起着重要的作用。这项提案中研究的问题涉及与密码学感兴趣的不变量密切相关的不变量，并可能对它们产生一些洞察。

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.