Eigenvarieties

目前，我们正在见证经典朗兰兹计划范围的一次重要而自然的扩大。这一新的数学发展利用了模形式的傅立叶系数之间的丰富的同余结构，以及更一般的自同构表示，将无限多个原本完全不同的自同构表示连接到有限维参数空间中。根据一项统计，目前似乎正在进行六个独立的、基本上同时进行的、依附于代数群的自同构形式的参数化(p-进)空间及其伴随的伽罗瓦表示的构造。这些参数空间被称为“特征变差”或“黑克变差”，它们是由不同的人在不同但有时是重叠的背景下构造的：对于高阶酉群，对于高阶辛群，对于数域上的一般线性群。本征簇是数论、代数几何、解析几何(主要是p进的)和群表示理论(包括自同构表示和伽罗瓦表示)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。哈佛大学2006年春季学期的本征品种项目旨在将从事这些建筑工作的许多人聚集在一起，提供关于这些材料的密集研究生课程和卫星研讨会。Ramanujan的经典工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的同余。这些同余表明，一种神秘的连贯性隐藏在大量基本算术现象的基础上，例如可以将N个对象的集合划分为子集合的方法的数量，或者给定某个欧几里德空间中的格子，最接近给定点的格点的数量，或者{\it模素数的多项式方程组的解的数量}。一张非凡的同余网络就像一种虚拟的粘合剂，将这些问题捆绑在一起。在接下来的几年里，寻找具有算术应用的同余，统一表示理论和模形式理论，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，它在戏剧性地证明有理数上椭圆曲线的模性方面发挥了作用。其中一家现在正处于这家企业重大扩张的边缘。希望哈佛大学2006年春季学期的本征变种计划将提供一个环境，在那里可以取得进一步的进展，建立对当前知识状况的连贯描述，并使研究生、博士后和其他感兴趣的数学家能够掌握这些新的发展。

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.