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Lattices with many Borcherds products


Authors: Jan Hendrik Bruinier, Stephan Ehlen and Eberhard Freitag
Journal: Math. Comp. 85 (2016), 1953-1981
MSC (2010): Primary 11F12, 11E20, 14C22
DOI: https://doi.org/10.1090/mcom/3059
Published electronically: November 9, 2015
MathSciNet review: 3471115
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Abstract: We prove that there are only finitely many isometry classes of even lattices $ L$ of signature $ (2,n)$ for which the space of cusp forms of weight $ 1+n/2$ for the Weil representation of the discriminant group of $ L$ is trivial. We compute the list of these lattices. They have the property that every Heegner divisor for the orthogonal group of $ L$ can be realized as the divisor of a Borcherds product. We obtain similar classification results in greater generality for finite quadratic modules.


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Additional Information

Jan Hendrik Bruinier
Affiliation: Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, D–64289 Darmstadt, Germany
Email: bruinier@mathematik.tu-darmstadt.de

Stephan Ehlen
Affiliation: Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, D–64289 Darmstadt, Germany
Email: ehlen@mathematik.tu-darmstadt.de

Eberhard Freitag
Affiliation: Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, D–69120 Heidelberg, Germany
Email: freitag@mathi.uni-heidelberg.de

DOI: https://doi.org/10.1090/mcom/3059
Received by editor(s): August 21, 2014
Received by editor(s) in revised form: February 6, 2015
Published electronically: November 9, 2015
Additional Notes: The first and the second authors were partially supported by DFG grant BR-2163/4-1.
Article copyright: © Copyright 2015 American Mathematical Society