Pullback of arithmetic theta series and its modularity for unitary Shimura curves

This paper is a complement of the modularity result of Bruinier, Howard, Kudla, Rapoport and Yang (BHKRY) for the special case U (1, 1) not considered there. The main idea to embed a U (1, 1) Shimura curve to many U (n-1, 1) Shimura varieties for big n, and prove a precise pullback formula of the generating series of arithmetic divisors. Afterwards, we use the modularity result of BHKRY together with existence of non-vanishing of classical theta series at any given point in the upper half plane to prove the modulartiy result on U (1, 1) Shimura curves.