Maximal parahoric arithmetic transfers, resolutions and modularity

2021/12/22 by Zhang, Zhiyu · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Advanced Mathematical Identities

paper · pdf · doi:10.48550/arxiv.2112.11994

Abstract

For any unramified quadratic extension of p-adic local fields F/F₀ (p>2), we formulate several arithmetic transfer conjectures at any maximal parahoric level, in the context of Zhang's relative trace formula approach to the arithmetic Gan--Gross--Prasad conjecture. The formulation involves a way to resolve the singularity of relevant moduli spaces via natural stratifications and modify derived fixed points. By a local-global method and double induction, we prove these conjectures for F₀ unramified over \mathbb Q_p, including the arithmetic fundamental lemma for p>2. Moreover, we prove new modularity results for arithmetic theta series at parahoric levels via a method of modification over \mathbb F_q and \mathbb C. Along the way, we study the complex and mod p geometry of Shimura varieties and special cycles. We introduce the relative Cayley map and also establish Jacquet--Rallis transfers at maximal parahoric levels.

Cited by

• A Proof of the Biquadratic Linear AFL for GL(4)

Related

• More regular formal moduli spaces and arithmetic transfer conjectures: the ramified quadratic case
• Quasi-canonical AFL and Arithmetic Transfer conjectures at parahoric levels
• Unitary Shimura varieties at ramified primes and arithmetic transfer
• On the Arithmetic Fundamental Lemma conjecture over a general p-adic field
• Kudla-Rapoport conjecture for unramified maximal parahoric level