  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.
