Arithmetic and Lifting of Automorphic Forms

研究者建议研究自同构形式和线性代数群理论中出现的算术问题，特别是在朗兰兹计划的背景下。作为一个例子，众所周知，在上半平面上的经典模形式的傅里叶系数中包含有丰富的算术信息。研究人员有一个正在进行的项目(与B.Gross和G.Savin)，以发展例外群G2上某些模形式的傅立叶系数理论，并从这些系数中提取算术信息。研究人员还对构造和分类还原群上的算术有趣的尖点形式感兴趣，并将它们与Arthur的某些有影响力的猜想联系起来。他打算利用特殊的theta对应来构造其中的一些尖点形式。自同构形的理论是现代数学研究的主要主题之一。除了本身是一门美丽的学科外，它还与数学的其他重要领域密切相关，例如数论(研究数字的性质，及其在密码学中的应用)和表示理论(研究对称性，及其在物理学中的应用)。通常，它在具体问题上有令人惊讶的应用，例如对通信和网络理论非常感兴趣的Ramanujan图的构造。在自转形式理论中，特别重要和难以捉摸的是某些被称为尖点形式的原始物体。研究者希望通过给出其中一些对象的明确结构来加深我们对这些基本对象的理解。

The investigator proposes to study arithmetic questions arising in the theory of automorphic forms and linear algebraic groups, especially in the context of the Langlands program. As an example, it is well-known that a wealth of arithmetic information is contained in the Fourier coefficients of a classical modular form on the upper half plane. The investigator has an ongoing project (with B. Gross and G. Savin) to develope a theory of Fourier coefficients for certain modular forms on the exceptional group G2 and to extract arithmetic information from these coefficients. The investigator is also interested in constructing and classifying arithmetically interesting cusp forms on reductive groups, and relating them to certain influential conjectures of Arthur. He intends to exploit the exceptional theta correspondence for the construction of some of these cusp forms.The theory of automorphic forms is one of the main themes of modern mathematical research. Besides being a beautiful subject in its own right, it is intimately connected with other important areas of mathematics such as number theory (the study of properties of numbers, with applications to cryptography)and representation theory (the study of symmetries, with applications to physics). Often, it has surprising applications to concrete problems, such as the construction of Ramanujan graphs which are of great interestin communication and network theory. Particularly important and elusive in the theory of automrophic forms are certain primitive objects known as cusp forms. The investigator hopes to further our understanding of these basic objects by giving explicit constructions of some of them.