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 conjecture 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.