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Holomorphic Borcherds products of singular weight for simple lattices of arbitrary level


Authors: Sebastian Opitz and Markus Schwagenscheidt
Journal: Proc. Amer. Math. Soc. 147 (2019), 4639-4653
MSC (2010): Primary 11F27
DOI: https://doi.org/10.1090/proc/14650
Published electronically: June 10, 2019
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Abstract: We classify the holomorphic Borcherds products of singular weight for all simple lattices of signature $ (2,n)$ with $ n \geq 3$. In addition to the automorphic products of singular weight for the simple lattices of square free level found by Dittmann, Hagemeier, and the second author, we obtain several automorphic products of singular weight $ 1/2$ for simple lattices of signature $ (2,3)$. We interpret them as Siegel modular forms of genus $ 2$ and explicitly describe them in terms of the ten even theta constants. In order to rule out further holomorphic Borcherds products of singular weight, we derive estimates for the Fourier coefficients of vector valued Eisenstein series, which are of independent interest.


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Additional Information

Sebastian Opitz
Affiliation: Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, 64289 Darmstadt, Germany
Email: opitz@mathematik.tu-darmstadt.de

Markus Schwagenscheidt
Affiliation: Department of Mathematics, University of Cologne, Weyertal 86–90, 50931 Cologne, Germany
Email: mschwage@math.uni-koeln.de

DOI: https://doi.org/10.1090/proc/14650
Received by editor(s): November 5, 2018
Received by editor(s) in revised form: February 13, 2019
Published electronically: June 10, 2019
Additional Notes: The first author was supported by the DFG Research Unit FOR 1920 “Symmetry, Geometry and Arithmetic” and the LOEWE Reseach Unit USAG
The second author was supported by the SFB-TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics”, funded by the DFG
Communicated by: Amanda Folsom
Article copyright: © Copyright 2019 American Mathematical Society