Transfer of Siegel cusp forms of degree 2
The holomorphic Siegel modular forms of degree two may be viewed as automorphic forms on the group \(\mathrm{GSp}(4)\). According to Langlands' principle of functoriality, one should be able to lift the automorphic representation so obtained to an automorphic representation on \(\mathrm{GL}(4)\) (or \(\mathrm{GL}(5)\)). This paper addresses the existence of these liftings, among others, for a holomorphic Siegel modular form \(F\) of degree two of full level that is not a Saito-Kurokawa lift from \(\mathrm{SL}(2)\).NEWLINENEWLINEThe proof is based on the converse theorem of \textit{J. W. Cogdell} and \textit{I. I. Piatetski-Shapiro} [Math. Res. Lett. 3, No. 1, 67--76 (1996; Zbl 0864.22009)]. It boils down to prove the niceness of Rankin-Selberg \(L\)-functions \(L(s, \pi \times \tau)\), where \(\pi\) is the automorphic representation generated by \(F\) and \(\tau\) stands for a cuspidal automorphic representation of \(\mathrm{GL}(2)\). Note that \(\pi\) is never generic, thus the Langlands-Shahidi method is not applicable here. One then makes use of \textit{M. Furusawa}'s integral representation for \(L(s, \pi \times \tau)\) [J. Reine Angew. Math. 438, 187--218 (1993; Zbl 0770.11025)], which unfolds to local integrals involving Whittaker functionals and Bessel functionals of \(\pi\). This requires a careful choice of local data and intricate computations. Also, a ``pull-back formula'' is applied to establish the entireness of \(L(s, \pi \times \tau)\).NEWLINENEWLINEApart from the existence of liftings, the authors also deduce results on the arithmeticity of critical \(L\)-values that seem out of reach of the techniques based on trace formula.
scientific article; zbMATH DE number 6357636
scientific article