p-adic automorphic forms, p-adic L-functions and Galois representations

p 进自守形式、p 进 L 函数和伽罗瓦表示

• 批准号: 1101229
• 负责人: Eric Jean-Paul Urban
• 金额: $ 18万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101229&HistoricalAwards=false
• 关键词: adic; automorphic; forms; functions; Galois

The domain of research of this project is the arithmetic of p-adic automorphic forms, their Galois representations and L-functions. Classical automorphic forms for a group G are functions on the adelic points of a reductive group G that satisfy nice transformation properties. Their arithmetic theory has yielded significant advances in the past few years:a proof Fermat's Last Theorem and a proof the Sato-Tate conjecture, to name two. The p-adic notions alluded above is a build-in concept to study congruences between these classical objects. Urban's proposal is a continuation on his work related to the construction and the study of congruences between Eisenstein series and cuspidal automorphic forms of various weights and levels and their links with p-adic L-functions and certain arithmetically defined groups called Selmer groups. Urban proposes to continue to build some of the foundations of the general theory of the p-adic automorphic forms and p-adic Eisenstein series with an eye one the important applications that will result. In particular, this theory applied to the case of unitary and symplectic groups will have important applications to the so-called p-adic Bloch-Kato conjecture. Here is a list of several topics that this project will deal with:(1) Dimension of irreducible components of Eigenvarieties(2) Galois representations for GL(n) and torsion classes for the cohomology of U(n,n),(3) p-adic deformations of holomorphic and nearly holomorphic automorphic forms,(4) Construction of p-adic measures attached to L-functions and Eisenstein series,(5) p-adic Euler system and p-adic L-functions. This project will enhance our knowledge of the deep relationships between p-adic automorphic forms, Galois representations, and their p-adic L-functions- a central focus of number theory - as well as have significant consequences for our understanding of mathematics in general. Urban intends to write a book on the general theory of p-adic automorphic representations to report on the many developments that has known the past decade. Half of the book will be devoted to the theory for general reductive group. In the second half, he will describe some applications with examples and propose directions of future research. In particular, important conjectures of the theory will be described in detail. This book will be intended mainly to graduate student and researchers. Urban works jointly with some PhD students and recent post-doctoral mathematicians on some of his projects by organizing seminar and workshops. He also wants to enhance the training of these young mathematicians and facilitate interactions with highly qualified mathematicians he will invite on a regular basis. Such an activity, his book project and the participation by Urban and his students in other seminars throughout the country and in international conferences will be very useful for the dissemination of the advances resulting from this project.

该项目的研究领域是P-ADIC汽车形式，其GALOIS表示和L功能的算术。 G组的经典自身形式是还满足良好转化属性的还原组G的adelic点的函数。在过去的几年中，他们的算术理论取得了重大进步：证明费马特的最后一个定理和证明了萨托·泰特（Sato-Tate）猜想的证明，仅为两个。上面提到的P-ADIC概念是研究这些经典对象之间的一致性的建筑概念。 Urban的提议是他的工作的延续，与艾森斯坦系列和各种权重和水平的Cuspidal自态形式之间的一致性有关，及其与P-ADIC L功能以及某些算术定义的组称为Selmer群体的联系。 Urban建议继续建立P-ADIC汽车形式的一般理论的一些基础和P-ADIC EISENSTEIN系列，这是将会导致的重要应用。特别是，该理论适用于单一和符号群体的情况，将在所谓的P-Adic Bloch-Kato猜想中具有重要的应用。 Here is a list of several topics that this project will deal with:(1) Dimension of irreducible components of Eigenvarieties(2) Galois representations for GL(n) and torsion classes for the cohomology of U(n,n),(3) p-adic deformations of holomorphic and nearly holomorphic automorphic forms,(4) Construction of p-adic measures attached to L-functions and Eisenstein系列，（5）P-ADIC EULER系统和P-Adic L功能。该项目将增强我们对P-ADIC自动形式，GALOIS表示及其P-Adic L功能之间的深厚关系的了解 - 数字理论的主要重点 - 以及对我们对数学总体学的理解产生重大影响。 Urban打算写一本关于P-ADIC汽车代表的一般理论的书，以报告过去十年来的许多发展。这本书的一半将专门用于一般还原群体的理论。下半年，他将用示例描述一些应用程序，并提出未来研究的指示。特别是，将详细描述该理论的重要猜想。这本书将主要旨在研究生和研究人员。 Urban通过组织研讨会和讲习班共同与一些博士生和最近的博士后数学家合作，在他的一些项目上合作。他还希望加强对这些年轻数学家的培训，并促进与高素质的数学家的互动，他将定期邀请。这样的活动，他的书籍项目以及Urban及其学生在全国和国际会议上的其他研讨会的参与对于传播该项目的进步非常有用。