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Level-raising for Saito–Kurokawa forms

Part of: Forms and linear algebraic groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 July 2009

Claus M. Sorensen
Claus M. Sorensen*
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA (email: claus@princeton.edu)
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Abstract

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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.

Keywords

automorphic formslevel-raising congruencestrace formulaSelmer groups

MSC classification

Secondary: 11E57: Classical groups 11E72: Galois cohomology of linear algebraic groups 11F27: Theta series; Weil representation; theta correspondences 11F33: Congruences for modular and $p$-adic modular forms 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F72: Spectral theory; Selberg trace formula

Information

Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 4 , July 2009 , pp. 915 - 953
Copyright
Copyright © Foundation Compositio Mathematica 2009

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