Fourier-Jacobi cycles and arithmetic relative trace formula (with an appendix by Chao Li and Yihang Zhu)

Abstract

In this article, we develop an arithmetic analogue of Fourier--Jacobi period integrals for a pair of unitary groups of equal rank. We construct the so-called Fourier--Jacobi cycles, which are algebraic cycles on the product of unitary Shimura varieties and abelian varieties. We propose the arithmetic Gan--Gross--Prasad conjecture for these cycles, which is related to central derivatives of certain Rankin--Selberg $L$-functions, and develop a relative trace formula approach toward this conjecture. As a necessary ingredient, we propose the conjecture of the corresponding arithmetic fundamental lemma, and confirm it for unitary groups of rank at most two and for the minuscule case.

arXiv commentv2: minor revision, 89 pages, published on Cambridge Journal of Mathematics

References in corpus (8)

• Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups
• Local-global compatibility and the action of monodromy on nearby cycles
• The fundamental lemma of Jacquet-Rallis in positive characteristics
• On Albanese torsors and the elementary obstruction
• On the Arithmetic Fundamental Lemma in the minuscule case
• On the arithmetic transfer conjecture for exotic smooth formal moduli spaces
• Fine Deligne-Lusztig varieties and Arithmetic Fundamental Lemmas
• Irreducibility of Theta Lifting for Unitary Groups