Families of p-adic modular forms

p-进模形式族

• 批准号: 0701048
• 负责人: Francesco Calegari
• 金额: $ 19.38万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701048&HistoricalAwards=false
• 关键词: Families; adic; modular; forms

Abstract for the proposal of Calegari DMS-0701048This project investigates the relationship between Galois representations and automorphic forms using tools from homology, commutative algebra, and p-adic analysis. Of particular interest are modular forms that are of cohomological type but are not associated to Shimura varieties, especially modular forms over an imaginary quadratic field K. In this case, the associated symmetric space quotients are hyperbolic manifolds of real dimension 3, and thus, the study of such forms is not amenable to the usual techniques of algebraic geometry. A long-term goal of the project is to establish the modularity of elliptic curves over K, adapting the method of Taylor-Wiles to this setting. Modular forms over K can be thought of geometrically as cohomology classes of certain local systems on arithmetic 3-manifolds. From a topological viewpoint, the cohomology of arithmetic 3-manifolds has been intensely studied in relation to such questions as Thurston's "Virtual Positive Betti Number Conjecture." However, in number theory, these cohomology classes have conjectural associations to motives and Galois representations. The tension between these different perspectives makes this a fertile area for interdisciplinary research.Wiles' famous work on Fermat's last theorem established a correspondence between two classes of disparate objects: elliptic curves, given by polynomial equations of degree three in two variables, and modular forms.Associated to each elliptic curve is a modular form, which is like the DNA of the corresponding elliptic curve: many properties of the elliptic curve can be determined directly from the modular form. An emerging theme in number theory (and beyond) in the last twenty years is the Langlands program, a vast generalization of this correspondence. In the Langlands program, one considers algebraic varieties, which are finite collections of polynomial equations of arbitrary degree, and automorphic forms, which, like modular forms in the classical case, play the role of algebraic variety DNA. For example, given a set of equations, one would like to know if there exist infinitely many solutions where all the variables are integers. Conjecturally, one can determine this directly from the corresponding automorphic form by computing whether a particular integral vanishes. The Langlands program is still, in many respects, in its infancy --- we neither know exactly how to relate systems of equations, in general, to automorphic forms, nor have we "cracked the code" for understanding how to extract useful information from automorphic forms, or have even determined all the components from which automorphic forms are built. Nevertheless, this broad area of study promises to be a guiding problem in number theory for the foreseeable future.

本文利用同调、交换代数和p进分析等工具研究伽罗瓦表示与自同构形式之间的关系。特别令人感兴趣的是上同调型的模形式，但与志村变体无关，特别是虚二次域k上的模形式。在这种情况下，相关的对称空间商是实维3的双曲流形，因此，这种形式的研究不适合代数几何的通常技术。该项目的长期目标是建立K上的椭圆曲线的模块化，使泰勒-怀尔斯方法适应于这种设置。K上的模形式可以被认为是算术3流形上某些局部系统的上同调类。从拓扑学的观点出发，对算术3流形的上同调进行了深入的研究，涉及到Thurston的“虚正Betti数猜想”等问题。然而，在数论中，这些上同类与动机和伽罗瓦表示有推测性的联系。这些不同观点之间的紧张关系使其成为跨学科研究的沃土。怀尔斯关于费马大定理的著名著作建立了两类完全不同的对象之间的对应关系：由两变量的三次多项式方程给出的椭圆曲线和模形式。与每条椭圆曲线相关联的是一个模形式，它就像对应椭圆曲线的DNA，可以直接从模形式确定椭圆曲线的许多性质。在过去的二十年里，数论（以及数论之外）的一个新兴主题是朗兰兹纲领，它是对这种对应性的广泛概括。在Langlands程序中，人们考虑代数变体，它们是任意次多项式方程的有限集合，以及自同构形式，它们与经典情况下的模形式一样，起着代数变体DNA的作用。例如，给定一组方程，人们想知道是否存在无穷多个解，其中所有变量都是整数。从推测上讲，我们可以通过计算一个特定的积分是否消失，直接从相应的自同构形式中确定这一点。在许多方面，朗兰兹纲领仍处于起步阶段——我们既不知道如何确切地将方程组与自同构形式联系起来，也没有“破解密码”来理解如何从自同构形式中提取有用的信息，甚至没有确定构建自同构形式的所有组成部分。然而，在可预见的未来，这一广泛的研究领域有望成为数论的指导问题。