On the Fourier--Jacobi coefficients
of certain Eisenstein series
for a unitary group
by Timothy J. Hickey
Doctoral Dissertation,
University of Chicago,
1986.
ABSTRACT (text-based)
The purpose of this dissertation is to develop an explicit formula for
the Fourier-Jacobi coefficients of certain adelic Eisenstein series on
the non-tube domain GU(2,1) by extending a technique used by Siegel,
Baily, Tsao, and Karel in the context of rational tube domains. This
work also relies on results of Shintani concerning the Fourier-Jacobi
coefficients of those modular forms on GU(2,1) which are simultaneous
eigenforms of the Hecke operators.
Our main theorem provides a formula for the Fourier-Jacobi
coefficients of certain Eisenstein series of weight k associated to
Hecke characters of the imaginary quadratic field K which is used to
provide a Q-structure for GU(2,1). The formula is deduced from an
expansion of the Eisenstein series which is closely related to the
Fourier-Jacobi expansion and which has the form of an infinite sum of
terms indexex by primitive adelic theta function. Each term is a
product of (1) an explicit monomial of Hecke and Dirichlet L-series
for the specified character and weight, and (2) a theta function
obtained by applying formal Dirichlet series of operators to the
indexed adelic theta function. The Dirichlet series consists of
operators on the graded ring of adelic theta functions and can be
expressed in terms of the eigenvalues of the Eisenstein series by a
result of Shintani. A consequence of this explicit formula is the
arithmeticity (in the sense of Shimura) of the Eisenstein series.