CAREER: Arithmetic of Cohomological Automorphic Forms

职业：上同调自同构形式的算术

• 批准号: 0846285
• 负责人: Francesco Calegari
• 金额: $ 40万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0846285&HistoricalAwards=false
• 关键词: CAREER; Arithmetic; Cohomological; Automorphic; Forms

The investigator studies the relationship between Galois representations and automorphic forms using tools from homology, commutative algebra, and group theory, in the context of the Langlands program. A major innovation in the investigator's recent research is the use of non-commutative Iwasawa theory to study p-adically completed cohomology associated to automorphic forms of cohomological type, in particular, those forms that are not necessarily associated to Shimura varieties. One such class of automorphic forms are modular forms over an imaginary quadratic field. 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. Modular forms over imaginary quadratic fields 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 continues to be a subject of intense study. The tension between the number theoretical and topological perspectives makes this a fertile area for interdisciplinary research. The ultimate goal of the investigator's research is to formulate and prove a general reciprocity statement relating all Galois representations to the cohomology of arithmetic groups, generalizing the reciprocity conjecture of Langlands. A more specific goal is to establish conditional modularity theorems over imaginary quadratic fields, adapting the method of Taylor--Wiles.The Langlands reciprocity conjecture predicts a one-to-one correspondence between two classes of disparate objects: Galois representations, which describe certain symmetries of algebraic numbers, and automorphic forms. Associated to each Galois representation is an automorphic form, which is like the DNA of the corresponding Galois representation: many properties of the Galois representation can be determined directly from the automorphic form. One strategy for establishing this correspondence is a counting argument: show that the number of automorphic forms (of any fixedtype) equals the number of corresponding Galois representations. Wiles used this strategy to prove a special case of the Langlands reciprocity conjecture, from which Fermat's Last Theorem follows. Wiles' proof relies on certain auxiliary geometric constructions (Shimura varieties) that are not always available, and hence, any general argument requires a more robust method for counting automorphic forms. The investigator proposes such a method: by decomposing automorphic forms into their (mod-p) constituents and then gluing the pieces back together. This process (p-adic completion) relates the old counting problem to questions in topology, in particular, to well-known conjectures of Thurston. The investigator's research along these lines is helping to foster new collaborations between number theorists and low-dimensional topologists, which may lead to advances in several open questions in both fields. The investigator plans to organize several workshops for graduate students and postdocs designed to train young number theorists in this rapidly changing interdisciplinary field.

研究人员研究伽罗瓦表示和自守形式之间的关系，使用工具，从同源性，交换代数，群论，在朗兰兹计划的背景下。研究者最近研究的一个主要创新是使用非交换岩泽理论来研究与上同调类型的自守形式相关的p-adically完全上同调，特别是那些不一定与志村品种相关的形式。其中一类自守形式是虚二次域上的模形式。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。 虚二次域上的模形式可以被几何地认为是算术三维流形上某些局部系统的上同调类。从拓扑的观点来看，算术3-流形的上同调一直是一个深入研究的课题。数论和拓扑观点之间的紧张关系使其成为跨学科研究的沃土。研究人员的最终目标是制定和证明一个通用的互惠声明，将所有伽罗瓦表示与算术群的上同调联系起来，推广朗兰兹的互惠猜想。一个更具体的目标是建立虚二次域上的条件模定理，采用泰勒-怀尔斯的方法。朗兰兹互易猜想预测了两类不同对象之间的一一对应：伽罗瓦表示，描述代数数的某些对称性，以及自守形式。与每个伽罗瓦表示相关联的是一个自守形式，它就像相应伽罗瓦表示的DNA：伽罗瓦表示的许多性质可以直接从自守形式确定。建立这种对应的一个策略是计数论证：证明自守形式（任何固定类型）的数量等于相应的伽罗瓦表示的数量。怀尔斯用这种策略证明了朗兰兹互易猜想的一个特例，费马大定理就是从这个猜想推导出来的。怀尔斯的证明依赖于某些辅助的几何构造（志村变种），这些构造并不总是可用的，因此，任何一般的论证都需要一个更鲁棒的方法来计算自守形式。研究者提出了这样一种方法：通过将自守形式分解为它们的（mod-p）成分，然后将这些部分粘在一起。这个过程（p-adic完成）涉及到旧的计数问题的问题，在拓扑结构，特别是众所周知的arctures的瑟斯顿。研究人员的研究沿着这些路线是有助于促进数论学家和低维拓扑学家之间的新的合作，这可能会导致在两个领域的几个悬而未决的问题的进展。研究人员计划为研究生和博士后组织几次研讨会，旨在培养这个快速变化的跨学科领域的年轻数论家。