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

TL;DR

This paper develops an arithmetic analogue of Fourier-Jacobi period integrals, constructs algebraic cycles on Shimura varieties, and proposes a conjecture relating these cycles to derivatives of L-functions, using a trace formula approach.

Contribution

It introduces the concept of Fourier-Jacobi cycles on unitary Shimura varieties and formulates an arithmetic Gan--Gross--Prasad conjecture linking these cycles to L-function derivatives.

Findings

Proposed the arithmetic Gan--Gross--Prasad conjecture for Fourier-Jacobi cycles.

Confirmed the arithmetic fundamental lemma for rank at most two and minuscule cases.

Developed a relative trace formula approach to study these conjectures.

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.