P-adic Deformations of Eisenstein Series

爱森斯坦级数的 P 进变形

• 批准号: 0401131
• 负责人: Eric Jean-Paul Urban
• 金额: $ 14.1万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing 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=0401131&HistoricalAwards=false
• 关键词: adic; Deformations; Eisenstein; Series

Abstract of Eric Urban's award DMS-0401131. "p-adic deformation of Eisenstein series"Urban will pursue his research on the construction and the study of congruences between automorphic forms in order to attack some of the Bloch-Kato conjectures and the Iwasawa main conjectures relating critical L-values and the size of Selmer groups. He will especially work on such congruences between Eisenstein series and cusp forms as in the works by Mazur-Wiles or in his work on the main conjecture for the symmetric square of an ordinary elliptic curve. In that direction, the PI will continue his joint work in progress with C. Skinner proving the Iwasawa main conjecture for elliptic curves via the study of the Eisenstein ideal for U(2,2). He also plans to guide some Ph-D students on a similar topic for some other unitary groups. The PI will also work on the theory of p-adic families of automorphic forms for general reductive groups by both geometric and topological approaches. This work will be undertaken for its own interest but also because the study of such congruences for Eisenstein series has numerous powerful applications. By these means, Urban plans to obtain some new and general constructions of certain arithmetical fundamental objects as for instance $p$-adic Euler systems that will be an essential tool to bound above the size of some Selmer groups.Urban's research field is the arithmetic theory of automorphic forms. Those are certain holomorphic functions having many symmetries and whose deep arithmetical properties are read from their Fourier coefficients. These objects play a fundamental role in number theory as their study has been very fruitful in the last two decades, leading A. Wiles to the proof of the Fermat's last theorem. The ultimate goal of the inverstigator's research is to relate two apparently unrelated objects which are the special values of L-functions (a meromorphic function on the complex planes associated to an automorphic form) and the order of certain Selmer groups (a generalization of the class group of a number field) via the study of congruences between automorphic forms. One of the main tools that will be used in Urban's investigation is the theory of p-adic modular forms, a theory that encodes all the congruences between modular forms modulo arbitrary high powers of a given prime number p. He will develop new aspects of this theory for general reductive groups and apply it to relevant modular forms called Eisenstein series whose constant terms carry information on critical L-values. Among other things, the results of this research will also have some important implications to the p-adic and classical conjecture of Birch and Swinnerton-Dyer that are major conjectures on the set of solutions of cubic equations in two variables

Eric Urban的获奖摘要DMS-0401131。“爱森斯坦级数的p-adic变形“厄本将继续他的研究建设和自守形式之间的同余关系的研究，以攻击一些布洛赫-加藤定理和岩泽主要定理有关的临界L-值和大小的塞尔默群。 他将特别致力于这种同余关系之间的爱森斯坦系列和尖点形式的作品，由马祖尔，怀尔斯或在他的工作主要猜想的对称广场的一个普通的椭圆曲线。在这方面，PI将继续与C. Skinner通过研究U（2，2）的Eisenstein理想证明了椭圆曲线的Iwasawa主猜想。他还计划指导一些博士生对其他一些单一的群体类似的主题。 PI还将通过几何和拓扑方法研究一般还原群的自守形式的p进族理论。这项工作将进行自己的利益，但也因为研究这种同余式的爱森斯坦系列有许多强大的应用。通过这些手段，城市计划获得一些新的和一般建设的某些算术基本对象，例如$p$-进欧拉系统，这将是一个必不可少的工具，以约束以上的大小，一些塞尔默groups.Urban的研究领域是算术理论的自守形式。 这些是具有许多对称性的某些全纯函数，其深刻的算术性质可以从它们的傅立叶系数中读取。这些对象在数论中起着基础性的作用，因为它们的研究在过去的二十年里取得了丰硕的成果，导致A。费马最后定理的证明。研究者的最终目标是通过研究自守形式之间的同余关系，将两个显然无关的对象联系起来，这两个对象是L-函数（与自守形式相关的复平面上的亚纯函数）的特殊值和某些塞尔默群（数域的类群的推广）的阶。一个主要的工具，将用于城市的调查是理论的p进模块化形式，一个理论，编码之间的所有同余模块化形式模任意高权力的一个给定的素数p，他将开发新的方面，这一理论的一般还原群，并将其应用到相关的模块化形式称为爱森斯坦系列的常数项进行信息的关键L-值。除此之外，本文的研究结果对三次二元方程解集的主要理论--Birch和Swinnerton-Dyer的p-adic猜想和经典猜想也有重要的意义