This research will concentrate on the study of arithmetic aspects of automorphic forms on a unitary group G in three variables with respect to a CM extension E/F. Three related topics will be investigated. The first is an investigation of the coefficients in the Fourier-Jacobi expansion of a holomorphic automorphic form on G which is an eigenfunction of all Hecke operators. The coefficients, which are theta functions on an abelian variety with complex multiplication by E, are determined by the eigenvalues of the Hecke operators, but no explicit formula is known. It is suggested that the coefficients are related to special values of L-functions. The second topic concerns the study of automorphic forms on G and their Fourier- Jacobi expansions over rings of integers. The third topic is an investigation of the Galois representations associated to automorphic representations of G. This research will concentrate on those areas of number theory concerning automorphic forms of many variables and abelian varieties. The former are functions which encode much of the information on the number theoretic problem to be solved allowing one to use the powerful techniques of analysis in the solution. The latter are geometric objects which are relevant to the number theory problem. Together these objects form powerful tools to solve the very deep problems of number theory. This P.I. is a master of these tools and so much of great interest will result from this research.
