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In this article, we develop an arithmetic analogue of Fourier--Jacobi periodintegrals for a pair of unitary groups of equal rank. We construct theso-called Fourier--Jacobi cycles, which are algebraic cycles on the product ofunitary Shimura varieties and abelian varieties. We propose the arithmeticGan--Gross--Prasad conjecture for these cycles, which is related to centralderivatives of certain Rankin--Selberg $L$-functions, and develop a relativetrace formula approach toward this conjecture. As a necessary ingredient, wepropose the conjecture of the corresponding arithmetic fundamental lemma, andconfirm it for unitary groups of rank at most two and for the minuscule case.

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