Arithmetic and Lifting of Automorphic Forms

调查人员建议研究算术问题所产生的理论自守形式和线性代数群，特别是在范围内的朗兰兹计划。作为一个例子，众所周知，丰富的算术信息包含在上半平面上的经典模形式的傅立叶系数中。研究者有一个正在进行的项目（与B。Gross和G. Savin）发展了例外群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。研究者还对构造和分类约化群上的算术有趣尖点形式感兴趣，并将它们与亚瑟的某些有影响力的理论联系起来。他打算利用特殊的θ对应的建设一些这些尖点形式。理论的自守形式是一个主要的主题，现代数学研究。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，如数论（研究数字的性质，并应用于密码学）和表示论（研究对称性，并应用于物理学）。通常，它有令人惊讶的应用到具体的问题，如拉马努金图的建设是非常有趣的通信和网络理论。在自食形式理论中，特别重要和难以捉摸的是某些被称为尖点形式的原始对象。研究者希望通过给出其中一些对象的明确结构来进一步理解这些基本对象。

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.