Non-reductive special cycles and Twisted Arithmetic Fundamental Lemma

2024/06/03 by Zhang, Zhiyu
Mathematics · #11F67 #11G40 #14G35 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2406.00986

openalex publication_date 2024/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider arithmetic analogs of the relative Langlands program and applications of new non-reductive geometry. Firstly, we introduce mirabolic special cycles, which produce special cycles on many Hodge type Rapoport-Zink spaces via pullbacks e.g. Kudla--Rapoport cycles. Secondly, we formulate arithmetic intersection problems for these cycles and formulate a method of arithmetic induction. As a main example, we formulate arithmetic twisted Gan--Gross--Prasad conjectures on unitary Shimura varieties and prove a key twisted arithmetic fundamental lemma using mirabolic special cycles, arithmetic inductions, and Weil type relative trace formulas.

Related

• Non-reductive cycles and twisted arithmetic transfers for Shimura curves
• Arithmetic diagonal cycles on unitary Shimura varieties
• Refined Cycle Maps
• An arithmetic intersection conjecture
• Arithmetic special cycles and Jacobi forms