Eigenvarieties

目前，我们正目睹经典朗兰兹纲领范围的一个重要的、自然的扩展。 这种新的数学发展利用模形式的傅立叶系数之间的同余关系的丰富结构，更一般地说，利用自守表示，将无限多个不同的自守表示联系在一起，形成有限维参数空间。 据一项统计，似乎有六个独立的，基本上同时进行的建设，参数化（p-adic）空间的自守形式连接到代数群，及其伴随的伽罗瓦表示。这些参数空间被称为“特征簇”或“Hecke簇”，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。 本征变化是数论、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的应用中使用。的 2006年春季学期，哈佛大学的特征多样性项目旨在汇集许多从事这些建设的人，提供关于这种材料的密集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，发掘出了包含重要数论信息的惊人的同余式。这些同余式表明，一种神秘的一致性隐藏在大量基本算术现象的基础上，例如，你可以将N个对象的集合分成子集合的方式，或者在某些欧几里得空间中给定一个格，最接近给定点的格点的数量，或者多项式方程组模素数的解的数量。 一个非同寻常的一致性网络就像一种虚拟的胶水，将这些问题联系在一起。在此期间，寻找有算术应用的同余，统一表示论和模形式理论，指导了许多数论工作。 在过去的几十年里，这种探索直接参与了数论的许多重要进展。 例如，几年前，它在有理数上椭圆曲线的模性的戏剧性证明中发挥了作用。 一个人现在正处于这个企业的重大扩张的边缘。我们希望，哈佛大学2006年春季学期的本征多样性计划将提供一个环境，在那里可以取得进一步的进展，在那里将建立一个连贯的帐户，目前的知识状态，研究生，博士后和其他感兴趣的数学家，可以掌握这些新的发展。

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.