Families of p-adic modular forms

Calegari DMS-0701048提案摘要本项目使用同调、交换代数和p-adic分析的工具研究伽罗瓦表示和自守形式之间的关系。特别感兴趣的是上同调类型的模形式，但不与志村品种，特别是模形式在虚二次域K。在这种情况下，相关联的对称空间流形是真实的维度为3的双曲流形，因此，对这种形式的研究不符合代数几何的通常技术。该项目的一个长期目标是建立K上椭圆曲线的模块化，使泰勒-怀尔斯方法适应这种设置。K上的模形式在几何上可以被认为是算术三维流形上某些局部系统的上同调类。从拓扑的观点出发，算术三维流形的上同调已经被深入研究，并与Thurston的“虚正Betti数猜想”等问题密切相关。然而，在数论中，这些上同调类与动机和伽罗瓦表示有着内在的联系。这些不同观点之间的张力使得这一领域成为跨学科研究的沃土。怀尔斯关于费马最后定理的著名著作建立了两类不同对象之间的对应关系：椭圆曲线，由二元三次多项式方程给出，和模形式。与每条椭圆曲线相关联的是模形式，就像相应椭圆曲线的DNA：椭圆曲线的许多性质可以直接从模形式确定。在过去的二十年里，数论（以及其他领域）的一个新兴主题是朗兰兹纲领，它是对这种对应关系的一个广泛推广。在朗兰兹纲领中，人们考虑代数簇（任意次数的多项式方程的有限集合）和自守形式（就像经典情况下的模形式一样，扮演代数簇DNA的角色）。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。从猜想上讲，我们可以通过计算某个特定积分是否为零，直接从相应的自守形式中确定这一点。朗兰兹纲领在许多方面仍处于起步阶段-我们既不确切知道如何将方程组与自守形式联系起来，也没有“破解”如何从自守形式中提取有用信息的密码，甚至没有确定自守形式的所有组成部分。尽管如此，在可预见的未来，这一广泛的研究领域有望成为数论中的一个指导性问题。

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.