Eigenvarieties

特征簇

• 批准号: 0514066
• 负责人: Barry Mazur
• 金额: --
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-08-01 至 2007-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0514066&HistoricalAwards=false
• 关键词: Eigenvarieties

At present we are witnessing an important, and natural, expansion of the scope of the classical Langlands program. This new mathematical development makes use of the rich structure of congruences between Fourier coefficients of modular forms, and more generally of automorphic representations, to tie together infinitely many otherwise disparate automorphic representations into finite-dimensional parameter spaces. By one count, there seems to be six independent, essentially simultaneous constructions currently underway, of parametrized (p-adic) spaces of automorphic forms attached to algebraic groups, and their concomitant Galois representations. These parameter spaces are called ``eigenvarieties," or ``Hecke varieties," and are being constructed by different people, in different but sometimes overlapping contexts: for unitary groups of higher rank, for symplectic groups of high rank, for general linear groups over number fields. Eigenvarieties are a unifying force for classical and modern aspects of number theory, algebraic geometry, analytic geometry (p-adic, mainly) and the theory of group representations (both automorphic representations and Galois representations). Some of this work has already been used in important applications. The Eigenvarieties program at Harvard University during the Spring semester 2006 is intended to bring together many of the people working on these constructions to provide intensive graduate courses on this material and satellite seminars. The classical work of Ramanujan, that dealt with the arithmetic properties of the Fourier coefficients of modular forms, unearthed striking congruences that contain important number theoretic information. These congruences suggest that a mysterious coherence underlies a large assortment of basic arithmetic phenomena such as the number of ways you can separate a collection of N objects into subcollections, or given a lattice in some Euclidean space, the number of lattice points closest to a given point, or {\it the number of solutions of a system of polynomial equations modulo a prime number}. An extraordinary web of congruences acts as a virtual glue that binds such problems together. In the intervening years, the search for congruences that have arithmetic applications, that unify representation theory, and the theory of modular forms, has guided much number-theoretic work. This search has been directly involved in many of the important advances in number theory in the past few decades. For example, it played its role in the dramatic proof of modularity of elliptic curves over the rational numbers, a few years ago. One is now on the verge of a significant expansion of this enterprise. The hope is that the Eigenvarieties program at Harvard University during the Spring semester 2006 program will provide a milieu where further progress can be made, where a coherent account of the current state of knowledge will be established, and where graduate students, and also post-docs and other interested mathematicians, can gain mastery of these new developments.

目前，我们正在见证经典朗兰兹纲领范围的重要且自然的扩展。 这种新的数学发展利用模形式的傅里叶系数之间的丰富同余结构，以及更一般的自同构表示，将无限多个不同的自同构表示连接到有限维参数空间中。 据一项统计，目前似乎有六个独立的、本质上同时进行的构造，即附加于代数群的自守形式的参数化（p-adic）空间，以及它们伴随的伽罗瓦表示。这些参数空间被称为“特征变体”或“赫克变体”，由不同的人在不同但有时重叠的上下文中构造：对于高阶酉群，对于高阶辛群，对于数域上的一般线性群。 特征变量是数论、代数几何、解析几何（主要是 p-adic）和群表示理论（自守表示和伽罗瓦表示）的经典和现代方面的统一力量。其中一些工作已经在重要应用中使用。哈佛大学 2006 年春季学期的 Eigenvarieties 计划旨在将许多从事这些建筑工作的人员聚集在一起，提供有关该材料的强化研究生课程和卫星研讨会。 拉马努金的经典著作涉及模形式的傅里叶系数的算术性质，发现了包含重要数论信息的惊人同余。这些同余表明，大量基本算术现象背后隐藏着一种神秘的相干性，例如将 N 个对象的集合分成子集合的方法数量，或者给定某个欧几里得空间中的格，最接近给定点的格点的数量，或者{\it 多项式方程组以素数为模的解的数量}。 一个非凡的同余网就像虚拟的粘合剂一样，将这些问题粘合在一起。在随后的几年里，对具有算术应用、统一表示论和模形式理论的同余式的探索指导了许多数论工作。 这种搜索直接涉及过去几十年来数论中的许多重要进展。 例如，几年前，它在有理数上的椭圆曲线模性的戏剧性证明中发挥了作用。 该企业现在正处于大规模扩张的边缘。我们希望哈佛大学 2006 年春季学期的特征变量项目能够提供一个能够取得进一步进展的环境，能够对当前的知识状况建立一个连贯的描述，并且研究生、博士后和其他感兴趣的数学家能够掌握这些新的发展。