Vector valued formal Fourier-Jacobi series

Abstract

H. Aoki showed that any symmetric formal Fourier-Jacobi series for the symplectic group Sp_2(Z) is the Fourier-Jacobi expansion of a holomorphic Siegel modular form. We prove an analogous result for vector valued symmetric formal Fourier-Jacobi series, by combining Aoki's theorem with facts about vector valued modular forms. Recently, this result was also proved independently by M. Raum using a different approach. As an application, by means of work of W. Zhang, modularity results for special cycles of codimension 2 on Shimura varieties associated to orthogonal groups can be derived.