Vector valued formal Fourier-Jacobi series

Bruinier, Jan

doi:10.1090/S0002-9939-2014-12272-6

Vector valued formal Fourier-Jacobi series
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by Jan Hendrik Bruinier PDF

Proc. Amer. Math. Soc. 143 (2015), 505-512 Request permission

Abstract:

H. Aoki showed that any symmetric formal Fourier-Jacobi series for the symplectic group $\mathrm {Sp}_2(\mathbb {Z})$ 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.

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