Arithmetic level raising for certain quaternionic unitary Shimura variety

TL;DR

This paper proves an arithmetic level raising theorem for quaternionic unitary Shimura varieties, advancing understanding of special fibers and conjectures related to orthogonal groups and Galois representations.

Contribution

It establishes a new level raising result in the ramified case for quaternionic unitary Shimura varieties, linking to major conjectures in number theory.

Findings

Proves an arithmetic level raising theorem for symplectic groups of degree four.

Provides a description of the supersingular locus related to classical Siegel threefold.

Connects the result to the Beilinson-Bloch-Kato and Gan-Gross-Prasad conjectures.

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.