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，我们目睹了古典Langlands计划范围的重要而自然的扩展。 这种新的数学开发利用了模块化形式的傅立叶系数与更普遍的自动形式表示之间的丰富结构，将无限的许多原本不同的自动形态表示形式联系起来，与有限维度参数空间联系起来。 一项计数，目前似乎正在进行六个独立的，本质上的同时构造，这些构造是由代数组附加的自动形式的参数化（P-ADIC）空间及其伴随的Galois表示。这些参数空间称为``特征变量，''或``hecke品种''，并且是由不同但有时在不同但有时重叠的上下文中的不同人建造的：对于较高等级的单一组，对于较高等级的符号组，对于一般线性组的高等级群体，数字领域。 特征值是数字理论的经典和现代方面，代数几何，分析几何形状（主要是P-ADIC）和群体表示理论（自动形式表示和Galois表示）的统一力量。其中一些工作已经用于重要应用程序。 2006年春季学期期间，哈佛大学的特征华人计划旨在汇集许多从事这些建筑的人，以在此材料和卫星研讨会上提供密集的研究生课程。 Ramanujan的经典作品，涉及模块化形式的傅立叶系数的算术特性，其中包含重要数字理论信息的出土惊人的一致性。 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}. 非凡的一致性网络是将这些问题绑定在一起的虚拟胶。在随后的几年中，寻找具有算术应用，统一表示理论和模块化形式理论的一致性指导了许多理论工作。 在过去的几十年中，这种搜索直接参与了数量理论的许多重要进步。 例如，几年前，它在椭圆曲线模块化的戏剧性证明中发挥了作用。 现在，人们正处于该企业的重大扩张的边缘。希望在2006年春季学期的计划期间，哈佛大学的特征值计划将提供一个环境，在这里可以进一步取得进一步的进步，在这些过程中建立了当前知识状态，研究生以及其他感兴趣的数学家可以掌握这些新发展。