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Level-raising for Saito–Kurokawa forms
| Content Provider | Scilit |
|---|---|
| Author | Sorensen, Claus M. |
| Copyright Year | 2009 |
| Description | Journal: Compositio Mathematica 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. |
| Ending Page | 953 |
| Starting Page | 915 |
| ISSN | 00221295 |
| e-ISSN | 15705846 |
| DOI | 10.1112/s0010437x09004084 |
| Journal | Compositio Mathematica |
| Issue Number | 4 |
| Volume Number | 145 |
| Language | English |
| Publisher | Wiley-Blackwell |
| Publisher Date | 2009-07-01 |
| Access Restriction | Open |
| Subject Keyword | Journal: Compositio Mathematica Mathematical Physics Quadratic Form Automorphic Forms Modular Form |
| Content Type | Text |
| Resource Type | Article |
| Subject | Algebra and Number Theory |
