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

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- by Sebastian Opitz and Markus Schwagenscheidt PDF
- Proc. Amer. Math. Soc.
**147**(2019), 4639-4653 Request permission

## 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.## References

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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
- MR Author ID: 1094068
- Email: mschwage@math.uni-koeln.de
- 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
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**147**(2019), 4639-4653 - MSC (2010): Primary 11F27
- DOI: https://doi.org/10.1090/proc/14650
- MathSciNet review: 4011501