Eigenvarieties

目前，我们正在目睹经典朗兰兹纲领范围的重要的、自然的扩大。这个新的数学发展利用了模形式的傅里叶系数之间的丰富的同余结构，以及更普遍的自同构表示，将无限多个其他不同的自同构表示结合到有限维参数空间中。通过一次计数，目前似乎有六个独立的，本质上是同时进行的构造，连接到代数群的自同构形式的参数化（p进）空间，以及它们伴随的伽罗瓦表示。这些参数空间被称为“特征变异”或“赫克变异”，由不同的人在不同但有时重叠的情况下构造：对于高秩的酉群，对于高秩的辛群，对于数域上的一般线性群。特征变分是数论、代数几何、解析几何（主要是p进几何）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得到了应用。哈佛大学2006年春季学期的特征变体计划旨在将从事这些结构的许多人聚集在一起，提供有关这些材料和卫星研讨会的密集研究生课程。拉马努金的经典著作处理了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，一种神秘的相干性是大量基本算术现象的基础，例如将N个对象的集合分成子集合的方法的数量，或者在某个欧几里得空间中给定一个晶格，最接近给定点的晶格点的数量，或者是多项式方程系统模取素数的解的数量。一个非凡的一致性网络就像虚拟的胶水，把这些问题粘合在一起。在这中间的几年里，对具有算术应用、统一表示理论和模形式理论的同余的探索，指导了许多数论工作。在过去的几十年里，这种研究直接涉及到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。一个公司现在正处于这个企业大规模扩张的边缘。我们希望哈佛大学在2006年春季学期的特征变量项目将提供一个环境，在那里可以取得进一步的进展，在那里可以建立对当前知识状态的连贯描述，在那里研究生，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.