Families of p-adic modular forms

摘要根据Calegari DMS-0701048的建议，本项目利用同调、交换代数和p-进分析等工具研究了伽罗瓦表示与自同构形之间的关系。特别令人感兴趣的是上同调型但与Shimura簇无关的模形式，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数维3的双曲流形，因此，这种形式的研究不服从于通常的代数几何技巧。该项目的一个长期目标是建立K上椭圆曲线的模性，使Taylor-Wiles方法适应这种设置。K上的模形式在几何上可以看作是算术3-流形上某些局部系统的上同调类。从拓扑学的观点来看，算术3-流形的上同调与瑟斯顿的“虚正Betti数猜想”等问题有密切的关系。然而，在数论中，这些上同调类与动机和伽罗瓦表示有猜想的联系。这些不同视角之间的张力使这一领域成为跨学科研究的肥沃领域。Wiles关于费马最后定理的著名工作在两类不同的对象之间建立了对应：椭圆曲线，由两个变量的三次多项式给出，以及模形式。与每条椭圆曲线相关联的是模形式，这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(及以后)的一个新兴主题是朗兰兹计划，它是这种对应关系的广泛推广。在朗兰兹程序中，人们考虑了代数簇和自同构形，代数簇是任意次多项式方程的有限集合，而自同构形就像经典情况下的模形式一样，扮演着代数簇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所有变量都是整数的解。推测，可以通过计算特定积分是否为零，直接从相应的自同构形式确定这一点。朗兰兹计划在许多方面仍然处于初级阶段-我们既不知道如何准确地将一般的方程系统与自同构形式联系起来，也没有为理解如何从自同构形式中提取有用信息而“破解代码”，甚至还没有确定建立自同构形式的所有组件。然而，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.