Generalizing the Gross-Zagier formula for Rankin L-vaues and heights of CM points, the arithmetic version of a conjecture of Gan-Gross-Prasad relates some L-values and heights on Shimura varieties. I'll describe a relative-trace-formula-like approach to the conjecture. One question arising from this approach is an identity (called "arithmetic fundamental lemma" or AFL) between intersection numbers on unitary Rapoport-Zink space and relative orbital integrals. The lower rank cases were proved earlier; recent progress on the AFL has been made in a joint work with Rapoport and Terstiege.