Arithmetic level raising for certain quaternionic unitary Shimura variety

2022/04/14 by Haining Wang, Wang, Haining
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2204.06976

openalex publication_date 2022/04/14 · openalex created_date 2022/04/26 · openalex updated_date 2026/07/28

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.

Related

• Deformation of rigid Galois representations and cohomology of certain quaternionic unitary Shimura variety
• On Shimura varieties for unitary groups
• On the Bloch-Kato conjecture for some four-dimensional symplectic Galois representations
• Motivic cohomology of quaternionic Shimura varieties and level raising
• Arithmetic diagonal cycles on unitary Shimura varieties