CAREER: Arithmetic of Cohomological Automorphic Forms

研究者研究伽罗瓦表示和自同构形式之间的关系，使用同调，交换代数和群论的工具，在朗兰兹程序的背景下。研究者最近研究的一个主要创新是利用非交换Iwasawa理论研究了与上同型的自同构形式相关的p-完全上同，特别是那些与志村变体无关的形式。其中一类自同构形式是虚二次域上的模形式。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。虚二次域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流形的上同调一直是一个研究热点。数论和拓扑学观点之间的紧张关系使这一领域成为跨学科研究的沃土。研究者研究的最终目标是建立并证明一个将所有伽罗瓦表示与算术群上同调联系起来的一般互易命题，推广朗兰兹的互易猜想。一个更具体的目标是在虚二次域上建立条件模块化定理，采用泰勒-怀尔斯的方法。朗兰兹互易猜想预测了两类完全不同的对象之间的一对一对应关系：伽罗瓦表示，它描述了代数数的某些对称性，以及自同构形式。与每个伽罗瓦表示相关联的是一个自同构形式，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同构形式确定。建立这种对应关系的一个策略是计数论证：证明（任何固定类型的）自同构形式的数量等于相应伽罗瓦表示的数量。怀尔斯用这种方法证明了朗兰兹互易猜想的一个特例，而费马大定理就是从这个特例推导出来的。Wiles的证明依赖于某些并不总是可用的辅助几何结构（Shimura变体），因此，任何一般论证都需要一个更健壮的方法来计数自同构形式。研究者提出了这样一种方法：将自同构形式分解成它们的（mod-p）成分，然后将碎片粘合在一起。这个过程（p进补全）将旧的计数问题与拓扑学中的问题联系起来，特别是与著名的瑟斯顿猜想联系起来。研究者沿着这些思路进行的研究有助于促进数论学家和低维拓扑学家之间的新合作，这可能会在这两个领域的几个开放问题上取得进展。研究者计划为研究生和博士后组织几次研讨会，旨在培养年轻的数论学家在这个快速变化的跨学科领域。

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.