Level-raising for GSp(4)

Abstract

This thesis 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 $Pi$ of Saito-Kurokawa type, arising from classical modular forms $f in S_4(Gamma_0(N))$ of square-free level and root number $epsilon_f=-1$. We first transfer $Pi$ to a suitable inner form $G$ such that $G(R)$ is compact modulo its center. This is achieved by viewing $G$ as a similitude spin group of a definite quadratic form in five variables, and then $ heta$-lifting the whole Waldspurger packet for $widetilde{SL}(2)$ determined by $f$. Thereby we obtain an automorphic representation $pi$ of $G$. For the inner form we prove a precise level-raising result, inspired by the work of Bellaiche and Clozel, and relying on computations of Schmidt. Thus we obtain a $ ilde{pi}$ congruent to $pi$, with a local component that is irreducibly induced from an unramified twist of the Steinberg representation of the Klingen Levi subgroup. To transfer $ ilde{pi}$ back to $GSp(4)$, we use Arthur's stable trace formula and the exhaustive work of Hales on Shalika germs and the fundamental lemma in this case. Since $ ilde{pi}$ has a local component of the above type, all endoscopic error terms vanish. Indeed, by Weissauer, we only need to show that such a component does not participate in the $ heta$-correspondence with any $GO(4)$. This is an exercise in using Kudla's filtration of the Jacquet modules of the Weil representation. Thus we get a cuspidal automorphic representation $ tilde{Pi}$ of $GSp(4)$ congruent to $Pi$, which is neither CAP nor endoscopic. In particular, its Galois representations are irreducible by work of Ramakrishnan. It is crucial for our application that we can arrange for $ ilde{Pi}$ to have vectors fixed by the non-special maximal compact subgroups at all primes dividing $N$. Since $G$ is necessarily ramified at some prime $r$, we have to show a non-special analogue of the fundamental lemma at $r$. Fortunately, by work of Kottwitz we can compare the involved orbital integrals to twisted orbital integrals over the unramified quadratic extension of $Q_r$. The inner form $G$ splits over this extension, and the comparison of the twisted orbital integrals can be done by hand. 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 dividing $N$, we construct a torsion class in the Selmer group of the motive $M_f(2)$.