Level-raising for Saito–Kurokawa forms

Abstract

AbstractThis paper provides congruences between unstable and stable automorphic forms for the symplectic similitude group GSp(4). More precisely, we raise the level of certain CAP representations Π arising from classical modular forms. We first transfer Π toπon a suitable inner formG; this is achieved byθ-lifting. Forπ, we prove a precise level-raising result that is inspired by the work of Bellaiche and Clozel and which relies on computations of Schmidt. We thus obtain a$\tilde {\pi }$congruent toπ, with a local component that is irreducibly induced from an unramified twist of the Steinberg representation of the Klingen parabolic. To transfer$\tilde {\pi }$back to GSp(4), we use Arthur’s stable trace formula. Since$\tilde {\pi }$has a local component of the above type, all endoscopic error terms vanish. Indeed, by results due to Weissauer, we only need to show that such a component does not participate in theθ-correspondence with any GO(4); this is an exercise in using Kudla’s filtration of the Jacquet modules of the Weil representation. We therefore obtain a cuspidal automorphic representation$\tilde {\Pi }$of GSp(4), congruent to Π, which is neither CAP nor endoscopic. It is crucial for our application that we can arrange for$\tilde {\Pi }$to have vectors fixed by the non-special maximal compact subgroups at all primes dividingN. SinceGis necessarily ramified at some primer, we have to show a non-special analogue of the fundamental lemma atr. Finally, we give an application of our main result to the Bloch–Kato conjecture, assuming a conjecture of Skinner and Urban on the rank of the monodromy operators at the primes dividingN.