On the Shimura correspondence for \(GSp(4)\)
If \(\tilde f\) is a holomorphic modular form of half-integral weight, Shimura showed that one can associate to \(\tilde f\) an automorphic form f of integral weight. Moreover, he showed that the L-series for \(f\) can be expressed as a Rankin-Selberg integral of \(\tilde f\) with a \(GL(2)\) theta series. In this paper an analogue of the Shimura construction for generic automorphic forms of half-integral weight on GSp(4) (\(4\times 4\) matrices) is given. Let \({\tilde \phi}\) be an automorphic form of half-integral weight on \(GSp(4)\). The Shimura correspondent \(\phi\) is the automorphic form on \(Sp(4)\cong O(3,2)\) arising from the dual pair \((\tilde Sp(4), O(3,2))\) (the tilde denotes the metaplectic double cover). The L-group of \(O(3,2)\) is \(Sp(4,{\mathbb{C}})\), which has two standard representations, one of degree 4 and the other of degree 5. Here, for generic \({\tilde \phi}\) the degree 5 L-function \(L(s,\phi,5)\) is expressed as an integral involving \({\tilde \phi}\), and this is used to give its analytic continuation and functional equation. To give this integral representation, a metaplectic version of Novodvorsky's integral for the Rankin-Selberg convolution on \(GSp(4)\times GL(2)\) is used. Feeding a \(GL(2)\) theta series into this construction, one obtains a Dirichlet series summed over (Whittaker)-Fourier coefficients of \({\tilde \phi}\) indexed by squares. To compute this series, the Hecke operators on \(GSp(4)\) are computed. Provided \({\tilde \phi}\) is a Hecke eigenform, the Hecke relations imply that this Dirichlet series has an Euler product, and that at almost all places the local factor is the local component of \(L(s,\phi,5)\). Combining with the explicit functional equation for the necessary Eisenstein series, one obtains the analytic continuation and functional equation. The Shimura method has been used to obtain Euler products with functional equations in other metaplectic contexts, by Gelbart and Piatetski-Shapiro and by Bump and Hoffstein, in particular.
Novodvorsky's integral for the Rankin-Selberg convolution
scientific article; zbMATH DE number 4173087
scientific article