Holomorphic Borcherds products of singular weight for simple lattices of arbitrary level
HTML articles powered by AMS MathViewer
- by Sebastian Opitz and Markus Schwagenscheidt
- Proc. Amer. Math. Soc. 147 (2019), 4639-4653
- DOI: https://doi.org/10.1090/proc/14650
- Published electronically: June 10, 2019
- PDF | 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
- Richard E. Borcherds, Automorphic forms on $\textrm {O}_{s+2,2}(\textbf {R})$ and infinite products, Invent. Math. 120 (1995), no. 1, 161–213. MR 1323986, DOI 10.1007/BF01241126
- Richard E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), no. 3, 491–562. MR 1625724, DOI 10.1007/s002220050232
- Jan H. Bruinier, Borcherds products on O(2, $l$) and Chern classes of Heegner divisors, Lecture Notes in Mathematics, vol. 1780, Springer-Verlag, Berlin, 2002. MR 1903920, DOI 10.1007/b83278
- Jan Hendrik Bruinier, Stephan Ehlen, and Eberhard Freitag, Lattices with many Borcherds products, Math. Comp. 85 (2016), no. 300, 1953–1981. MR 3471115, DOI 10.1090/mcom/3059
- Jan Hendrik Bruinier, Stephan Ehlen, and Eberhard Freitag, Lattices with many Borcherds products, extended version, retrieved March 13, 2018, from https://github.com/sehlen/sfqm/blob/master/bruinier_ehlen_freitag_extended.pdf (2016).
- Jan Hendrik Bruinier and Michael Kuss, Eisenstein series attached to lattices and modular forms on orthogonal groups, Manuscripta Math. 106 (2001), no. 4, 443–459. MR 1875342, DOI 10.1007/s229-001-8027-1
- Jan Hendrik Bruinier and Martin Möller, Cones of Heegner divisors, to appear in J. Algebraic Geom. (2018).
- Michael Bundschuh, Über die Endlichkeit der Klassenzahl gerader Gitter der Signatur $(2,n)$ mit einfachem Kontrollraum, PhD Thesis, Heidelberg, 2001.
- Moritz Dittmann, Reflective automorphic products of squarefree level, Trans. Amer. Math. Soc., published electronically (2018), DOI:10.1090/tran/7620.
- Moritz Dittmann, Heike Hagemeier, and Markus Schwagenscheidt, Automorphic products of singular weight for simple lattices, Math. Z. 279 (2015), no. 1-2, 585–603. MR 3299869, DOI 10.1007/s00209-014-1383-6
- E. Freitag, Siegelsche Modulfunktionen, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 254, Springer-Verlag, Berlin, 1983 (German). MR 871067, DOI 10.1007/978-3-642-68649-8
- Valeri A. Gritsenko and Viacheslav V. Nikulin, Automorphic forms and Lorentzian Kac-Moody algebras. II, Internat. J. Math. 9 (1998), no. 2, 201–275. MR 1616929, DOI 10.1142/S0129167X98000117
- Stephen S. Kudla and TongHai Yang, Eisenstein series for SL(2), Sci. China Math. 53 (2010), no. 9, 2275–2316. MR 2718827, DOI 10.1007/s11425-010-4097-1
- Denis Lippolt, Thetanullwerte 2. Grades als Borcherds-Produkte, Diplomarbeit, Universität Heidelberg, 2008.
- Sebastian Opitz, Computation of Eisenstein series associated with discriminant forms, PhD thesis, Technische Universität Darmstadt, 2018.
- Nils R. Scheithauer, On the classification of automorphic products and generalized Kac-Moody algebras, Invent. Math. 164 (2006), no. 3, 641–678. MR 2221135, DOI 10.1007/s00222-006-0500-5
- Nils R. Scheithauer, Automorphic products of singular weight, Compos. Math. 153 (2017), no. 9, 1855–1892. MR 3705279, DOI 10.1112/S0010437X17007266
Bibliographic 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