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Vector valued formal Fourier-Jacobi series

Author: Jan Hendrik Bruinier
Journal: Proc. Amer. Math. Soc. 143 (2015), 505-512
MSC (2010): Primary 11F46, 11F50
Published electronically: September 19, 2014
MathSciNet review: 3283640
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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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Jan Hendrik Bruinier
Affiliation: Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstrasse 7, D–64289 Darmstadt, Germany

Received by editor(s): March 15, 2013
Received by editor(s) in revised form: May 16, 2013
Published electronically: September 19, 2014
Additional Notes: The author was partially supported by DFG grants BR-2163/2-2 and FOR 1920.
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2014 American Mathematical Society