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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 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 back to GSp(4), we use Arthur’s stable trace formula. Since 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 of GSp(4), congruent to Π, which is neither CAP nor endoscopic. It is crucial for our application that we can arrange for 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
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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