Vector valued formal Fourier-Jacobi series

TL;DR

This paper extends Aoki's theorem to vector valued symmetric formal Fourier-Jacobi series, showing they correspond to holomorphic Siegel modular forms, with implications for modularity in Shimura varieties.

Contribution

It proves an analogue of Aoki's theorem for vector valued series, combining existing results and methods to broaden the scope of modular form theory.

Findings

Vector valued symmetric formal Fourier-Jacobi series are Fourier-Jacobi expansions of holomorphic Siegel modular forms.

The result was independently proved by M. Raum using a different method.

Application to modularity of special cycles on Shimura varieties.

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.