Spinor L-Functions, Theta Correspondence, and Bessel Coefficients

The purpose of this note is to establish the entireness of spinor L-function of certain automorphic cuspidal representations of the group similitude symplectic group of order four over the rational numbers. Our study of spinor L-function is based on an integral representation which works only for generic representations. For this reason, while methods of this papers do not directly apply to the most interesting case of interest, ie. Siegel modular forms of genus two, they show what “should” be true for holomorphic forms; after all, generic forms are expected to be in a certain sense typical. These integrals which were introduced by M. Novodvorsky in the Corvallis conference [10] serve as one of the few available integral representations for the Spinor L-function of GSp(4). Some of the details missing in Novodvorsky’s original paper have been reproduced in Daniel Bump’s survey article [1]. Further details have been supplied by [19]. David Soudry has generalized the integrals considered in Novodvorsky’s paper to orthogonal groups of arbitrary odd degree. In light of the results of [19], it is sufficient to study the integral of Novodvorsky at the archimedean place. Archimedean computations are often forbidding, and unless one expects major simplifications due to the nature of the parameters, the resulting integrals are often quite hard to manage. In our case of interest, the work of Moriyama [12] benefits from exactly such simplifications when he treats the case of cuspidal representations with archimedean components in the generic (limit of) discrete series. In this work, we concentrate on those archimedean representations for which direct computations have yielded very little. For this reason, our methods are a bit indirect, in fact somewhat more indirect that what at first seems necessary. Our