Lattices with many Borcherds products

Bruinier, Jan; Ehlen, Stephan; Freitag, Eberhard

doi:10.1090/mcom/3059

Lattices with many Borcherds products
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by Jan Hendrik Bruinier, Stephan Ehlen and Eberhard Freitag PDF

Math. Comp. 85 (2016), 1953-1981 Request permission

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.

References

Irene A. Stegun (ed.), Pocketbook of mathematical functions, Verlag Harri Deutsch, Thun, 1984. Abridged edition of Handbook of mathematical functions edited by Milton Abramowitz and Irene A. Stegun; Material selected by Michael Danos and Johann Rafelski. MR 768931
Richard E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), no. 3, 491–562. MR 1625724, DOI 10.1007/s002220050232
Richard E. Borcherds, The Gross-Kohnen-Zagier theorem in higher dimensions, Duke Math. J. 97 (1999), no. 2, 219–233. MR 1682249, DOI 10.1215/S0012-7094-99-09710-7
Richard E. Borcherds, Reflection groups of Lorentzian lattices, Duke Math. J. 104 (2000), no. 2, 319–366. MR 1773561, DOI 10.1215/S0012-7094-00-10424-3
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, On the rank of Picard groups of modular varieties attached to orthogonal groups, Compositio Math. 133 (2002), no. 1, 49–63. MR 1918289, DOI 10.1023/A:1016357029843
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
M. Bunschuh, Über die Endlichkeit der Klassenzahl gerader Gitter der Signatur $(2,n)$ mit einfachem Kontrollraum, Dissertation universität Heidelberg (2002).
J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1999. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 1662447, DOI 10.1007/978-1-4757-6568-7
A. G. Earnest and J. S. Hsia, Spinor norms of local integral rotations. II, Pacific J. Math. 61 (1975), no. 1, 71–86. MR 404142
S. Ehlen, Finite quadratic modules and simple lattices, Source code and resources (2014). http://www.github.com/sehlen/sfqm.
Jürgen Fischer, An approach to the Selberg trace formula via the Selberg zeta-function, Lecture Notes in Mathematics, vol. 1253, Springer-Verlag, Berlin, 1987. MR 892317, DOI 10.1007/BFb0077696
E. Freitag, Riemann surfaces, CreateSpace Independent Publishing Platform (2014).
Gerard van der Geer, Hilbert modular surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 16, Springer-Verlag, Berlin, 1988. MR 930101, DOI 10.1007/978-3-642-61553-5
H. Hagemeier, Automorphe Produkte singulären Gewichts, Dissertation, Technische Universität Darmstadt (2010).
Yoshiyuki Kitaoka, Arithmetic of quadratic forms, Cambridge Tracts in Mathematics, vol. 106, Cambridge University Press, Cambridge, 1993. MR 1245266, DOI 10.1017/CBO9780511666155
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
V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111–177, 238 (Russian). MR 525944
Alexandre Nobs, Die irreduziblen Darstellungen der Gruppen $SL_{2}(Z_{p})$, insbesondere $SL_{2}(Z_{2})$. I, Comment. Math. Helv. 51 (1976), no. 4, 465–489 (German). MR 444787, DOI 10.1007/BF02568170
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
N. Scheithauer, Some constructions of modular forms for the Weil representation of $\operatorname {SL}_2(\mathbb {Z})$, Nagoya Math. J. 220 (2015), 1-43, DOI: 10.1215/00277630-3335405.
Nils-Peter Skoruppa, Über den Zusammenhang zwischen Jacobiformen und Modulformen halbganzen Gewichts, Bonner Mathematische Schriften [Bonn Mathematical Publications], vol. 159, Universität Bonn, Mathematisches Institut, Bonn, 1985 (German). Dissertation, Rheinische Friedrich-Wilhelms-Universität, Bonn, 1984. MR 806354
Nils-Peter Skoruppa, Jacobi forms of critical weight and Weil representations, Modular forms on Schiermonnikoog, Cambridge Univ. Press, Cambridge, 2008, pp. 239–266. MR 2512363, DOI 10.1017/CBO9780511543371.013
N.-P. Skoruppa, Finite Quadratic Modules and Weil representations, in preparation.
W. A. Stein et al. Sage Mathematics Software (Version 6.2). The Sage Development Team, 2014. http://www.sagemath.org.
Fredrik Strömberg, Weil representations associated with finite quadratic modules, Math. Z. 275 (2013), no. 1-2, 509–527. MR 3101818, DOI 10.1007/s00209-013-1145-x
G. L. Watson, Integral quadratic forms, Cambridge Tracts in Mathematics and Mathematical Physics, No. 51, Cambridge University Press, New York, 1960. MR 0118704
D. B. Zagier, Zetafunktionen und quadratische Körper, Hochschultext [University Textbooks], Springer-Verlag, Berlin-New York, 1981 (German). Eine Einführung in die höhere Zahlentheorie. [An introduction to higher number theory]. MR 631688