Arithmetic and Lifting of Automorphic Forms

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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.