Arithmetic level raising for certain quaternionic unitary Shimura variety

Abstract

In this article we prove an arithmetic level raising theorem for the symplectic group of degree four in the ramified case. This result is a key step towards the Beilinson-Bloch-Kato conjecture for certain Rankin-Selberg motives associated to orthogonal groups within the framework of the Gan-Gross-Prasad conjecture. The theorem itself can be also viewed as an analogue of the Ihara's lemma or the Tate conjectures for special fibers of Shimura varieties at ramified characteristics. The proof relies heavily on the description of the supersingular locus of certain quaternionic unitary Shimura variety which is closely related to the classical Siegel threefold. Contents 1. Introduction 1.1. Main results 1.2. Applications and prospect 1.3. Notations and conventions Acknowledgements 2. Quaternionic unitary Shimura variety 2.1. Quaternionic unitary groups 2.2. The local quaternionic unitary Shimura variety 2.3. The quaternionic unitary Shimura variety 3. Level raising conditions 4. Monodromy on the cohomology of the quaternionic unitary Shimura variety 4.1. Picard-Lefschetz formula 4.2. Descriptions of α and β 4.3. Nearby cycles on certain Shimura varieties 5. The level raising matrix 6. Computing the level raising matrix Key words and phrases. Shimura varieties, Arithmetic level raising, Vanishing cycles.HAINING WANG 6.1. A Hecke operator identity 30 6.2. Computing the level raising matrix 32 7. Tate cycles on the quaternionic unitary Shimura variety 37 7.1. Computing the supersingular matrix 38 7.2. Tate classes in the quaternionic unitary Shimura variety 40 8. Galois deformation rings and cohomology of Shimura varieties 43 8.1. Galois representation for GSp 4 43 8.2. Galois deformation rings 45 9. Arithmetic level raising and applications 47 9.1. Proof of Theorem 9.5 50 9.2. Level raising for GSp 4 54 References 55