We study the structures of Fourier coefficients of automorphic forms on symplectic groups based on their local and global structures related to Arthur parameters. This is a first step towards the general conjecture on the relation between the structure of Fourier coefficients and Arthur parameters for automorphic forms occurring in the discrete spectrum, given by the first named author.
In , J. Arthur classifies the automorphic discrete spectrum of symplectic groups up to global Arthur packets. We continue with our investigation of Fourier coefficients and their implication to the structure of the cuspidal spectrum for symplectic groups ( and ). As result, we obtain certain characterization and construction of small cuspidal automorphic representations and gain a better understanding of global Arthur packets and of the structure of local unramified components of the cuspidal spectrum, which has impacts to the generalized Ramanujan problem as posted by P. Sarnak in .
The existence of the well-known Jacquet–Langlands correspondence was established by
Jacquet and Langlands via the trace formula method in 1970. An explicit construction
of such a correspondence was obtained by Shimizu via theta series in 1972. In
this paper, we extend the automorphic descent method of Ginzburg–Rallis–Soudry
to a new setting. As a consequence, we recover the classical Jacquet–Langlands correspondence
for PGL(2) via a new explicit construction.