Motivic cohomology of quaternionic Shimura varieties and level raising

2019/01/07 by Zhou Rong, Zhou, Rong
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1901.01954

openalex publication_date 2019/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the motivic cohomology of the special fiber of quaternionic Shimura varieties at a prime of good reduction. We exhibit classes in these motivic cohomology groups and use this to give an explicit geometric realization of level raising between Hilbert modular forms. The main ingredient for our construction is a form of Ihara's Lemma for compact quaternionic Shimura surfaces which we prove by generalizing a method of Diamond-Taylor. Along the way we also verify the Hecke orbit conjecture for these quaternionic Shimura varieties which is a key input for our proof of Ihara's Lemma.

Related

• Arithmetic level raising for certain quaternionic unitary Shimura variety
• Motivic action for Siegel modular forms
• Towards an analogue of Ihara's lemma for Shimura curves
• Quantitative level lowering for weight two Hilbert modular forms
• Motivic cohomology of equicharacteristic schemes