SPECTRAL TRANSFER FOR METAPLECTIC GROUPS. I. LOCAL CHARACTER RELATIONS

Let $\widetilde{\text{Sp}}(2n)$ be the metaplectic covering of $\text{Sp}(2n)$ over a local field of characteristic zero. The core of the theory of endoscopy for $\widetilde{\text{Sp}}(2n)$ is the geometric transfer of orbital integrals to its elliptic endoscopic groups. The dual of this map, called the spectral transfer, is expected to yield endoscopic character relations which should reveal the internal structure of $L$-packets. As a first step, we characterize the image of the collective geometric transfer in the non-archimedean case, then reduce the spectral transfer to the case of cuspidal test functions by using a simple stable trace formula. In the archimedean case, we establish the character relations and determine the spectral transfer factors by rephrasing the works by Adams and Renard.

A stable trace formula for Igusa varieties

Igusa varieties are smooth varieties in positive characteristic p which are closely related to Shimura varieties and Rapoport–Zink spaces. One motivation for studying Igusa varieties is to analyse the representations in the cohomology of Shimura varieties which may be ramified at p. The main purpose of this work is to stabilize the trace formula for the cohomology of Igusa varieties arising from a PEL datum of type (A) or (C). Our proof is unconditional thanks to the recent proof of the fundamental lemma by Ngô, Waldspurger and many others.An earlier work of Kottwitz, which inspired our work and proves the stable trace formula for the special fibres of PEL Shimura varieties with good reduction, provides an explicit way to stabilize terms at ∞. Stabilization away from p and ∞ is carried out by the usual Langlands–Shelstad transfer as in work of Kottwitz. The key point of our work is to develop an explicit method to handle the orbital integrals at p. Our approach has the technical advantage that we do not need to deal with twisted orbital integrals or the twisted fundamental lemma.One application of our formula, among others, is the computation of the arithmetic cohomology of some compact PEL-type Shimura varieties of type (A) with non-trivial endoscopy. This is worked out in a preprint of the author's entitled ‘Galois representations arising from some compact Shimura varieties’.

Level-raising for Saito–Kurokawa forms

This 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 to π on a suitable inner form G; this is achieved by θ-lifting. For π, 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 to π, 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 the θ-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 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. 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.

Potential level-lowering for GSp(4)

In this article, we explore a beautiful idea of Skinner and Wiles in the context of GSp(4) over a totally real field. The main result provides congruences between automorphic forms which are Iwahori-spherical at a certain place ω, and forms with a tamely ramified principal series at ω, Thus, after base change to a finite solvable totally real extension, one can often lower the level at ω. For the proof, we first establish an analogue of the Jacquet–Langlands correspondence, using the stable trace formula. The congruences are then obtained on inner forms, which are compact at infinity modulo the centre, and split at all the finite places. The crucial ingredient allowing us to do so, is an important result of Roche on types for principal series representations of split reductive groups.