p-adic automorphic forms, p-adic L-functions, and Selmer groups

p 进自守形式、p 进 L 函数和 Selmer 群

• 批准号: 1407239
• 负责人: Eric Jean-Paul Urban
• 金额: $ 28.5万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1407239&HistoricalAwards=false
• 关键词: adic; automorphic; forms; functions; Selmer

This research project in number theory studies arithmetic properties of automorphic forms, which are functions that satisfy certain transformation properties. More specifically, the project is to study divisibility properties of these mathematical objects by high powers of a given prime number. The study of these properties has yielded significant advances in number theory in the past few years: a proof Fermat's Last Theorem, a proof the Sato-Tate conjecture, and a proof of the Iwasawa main conjecture for elliptic curves. The work resulting from this project will give more insight into some of the current most important problems in number theory. 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. The project will involve the training of graduate students.The domain of research of this project is the arithmetic of p-adic automorphic forms, their Galois representations, and L-functions. This project is a continuation of the PI's work related to the construction and the study of congruences between automorphic forms of various weights and levels and their links with p-adic L-functions and Selmer groups. The project will 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 and Birch and Swinnerton-Dyer conjecture. Studying the dimension of irreducible components of eigenvarieties, p-adic deformations of nearly over-convergent automorphic forms, critical p-adic L-functions, p-adic interpolation of automorphic periods, construction of p-adic measures attached to L-functions and Eisenstein series, p-adic Euler systems, and Kolyvagin systems and their link with p-adic L-functions are some of the topics that will be dealt with in this project.

该数论研究项目研究自守形式的算术性质，自守形式是满足某些变换性质的函数。更具体地说，该项目是通过给定素数的高幂来研究这些数学对象的整除性。对这些性质的研究在过去几年中在数论方面取得了重大进展：证明了费马大定理、证明了佐藤泰猜想以及证明了椭圆曲线的岩泽主猜想。该项目的工作将使人们更深入地了解数论中当前一些最重要的问题。该项目将增强我们对 p 进自守形式、伽罗瓦表示及其 p 进 L 函数（数论的核心焦点）之间深层关系的了解，并对我们对数学的总体理解产生重大影响。 该项目将涉及研究生的培训。该项目的研究领域是p-adic自守形式及其伽罗瓦表示和L函数的算术。该项目是 PI 工作的延续，该工作涉及构建和研究不同权重和级别的自同构形式之间的同余及其与 p 进 L 函数和 Selmer 群的联系。该项目将继续建立 p 进自守形式和 p 进爱森斯坦级数的一般理论的一些基础，并着眼于将产生的重要应用。特别是，该理论应用于酉群和辛群的情况，对于所谓的 p-adic Bloch-Kato 猜想以及 Birch 和 Swinnerton-Dyer 猜想将具有重要的应用。 研究特征变体的不可约分量的维数、近过度收敛自守形式的 p-adic 变形、临界 p-adic L 函数、自守周期的 p-adic 插值、附加于 L-函数和爱森斯坦级数的 p-adic 测度的构造、p-adic 欧拉系统和 Kolyvagin 系统及其与 p-adic 的联系 L 函数是本项目将讨论的一些主题。