Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-28T21:12:42.821Z Has data issue: false hasContentIssue false
  • English
  • Français

Computing Borcherds products

Part of: Computational number theory Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 August 2013

Dominic Gehre ,
Judith Kreuzer  and
Martin Raum
Dominic Gehre
Affiliation:
Lehrstuhl A für Mathematik,RWTH Aachen University,Templergraben 55, D-52056 Aachen,Germany email [email protected]@matha.rwth-aachen.de
Judith Kreuzer
Affiliation:
Lehrstuhl A für Mathematik,RWTH Aachen University,Templergraben 55, D-52056 Aachen,Germany email [email protected]@matha.rwth-aachen.de
Martin Raum
Affiliation:
ETH Mathematics Department,Rämistraße 101, CH-8092, Zürich,Switzerland email [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present an algorithm for computing Borcherds products, which has polynomial runtime. It deals efficiently with the bounds on Fourier expansion indices originating in Weyl chambers. Naive multiplication has exponential runtime due to inefficient handling of these bounds. An implementation of the new algorithm shows that it is also much faster in practice.

MSC classification

Secondary: 11F55: Other groups and their modular and automorphic forms (several variables) 11F30: Fourier coefficients of automorphic forms 11Y16: Algorithms; complexity 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 16 , October 2013 , pp. 200 - 215
Copyright
© The Author(s) 2013 

References

Bruinier, J. and Ono, K., ‘Heegner divisors, $L$ -functions and harmonic weak Maass forms’, Ann. of Math. (2) 172 (2010) 2135–2181.CrossRefGoogle Scholar
Borcherds, R., ‘Automorphic forms on ${O}_{s+ 2, 2} (R)$ and infinite products’, Invent. Math. 120 (1995) 161–213.CrossRefGoogle Scholar
Borcherds, R., ‘Automorphic forms with singularities on Grassmannians’, Invent. Math. 132 (1998) 491–562.CrossRefGoogle Scholar
Borcherds, R., ‘The Gross–Kohnen–Zagier theorem in higher dimensions’, Duke Math. J. 97 (1999) 219–233.CrossRefGoogle Scholar
Borcherds, R., ‘Correction to: The Gross–Kohnen–Zagier theorem in higher dimensions’, Duke Math. J. 105 (2000) 183–184.CrossRefGoogle Scholar
Braun, H., ‘Hermitian modular functions’, Ann. of Math. (2) 50 (1949) 827–855.CrossRefGoogle Scholar
Braun, H., ‘Hermitian modular functions. II’, Ann. of Math. (2) 51 (1950) 92–104.CrossRefGoogle Scholar
Braun, H., ‘Hermitian modular functions III’, Ann. of Math. (2) 53 (1951) 143–160.CrossRefGoogle Scholar
Bruinier, J., Borcherds products on O(2,l) and Chern classes of Heegner divisors, Lecture Notes in Mathematics 1780 (Springer, Berlin, 2002).CrossRefGoogle Scholar
Cléry, F. and Gritsenko, V., ‘Siegel modular forms of genus 2 with the simplest divisor’, Proc. Lond. Math. Soc. (3) 102 (2011) 1024–1052.CrossRefGoogle Scholar
Dern, T., ‘Hermitesche Modulformen zweiten Grades’, PhD Thesis, RWTH Aachen University, Germany, 2001.Google Scholar
Dabholkar, A. and Gaiotto, D., ‘Spectrum of CHL dyons from genus-two partition function’, J. High Energy Phys. 12 (2007) 087.CrossRefGoogle Scholar
Decker, W., Greuel, G., Pfister, G. and Schönemann, H., ‘Singular 3-1-1 — A computer algebra system for polynomial computations’, 2010, http://www.singular.uni-kl.de.Google Scholar
Dern, T. and Krieg, A., ‘Graded rings of Hermitian modular forms of degree 2’, Manuscripta Math. 110 (2003) 251–272.CrossRefGoogle Scholar
Dern, T. and Krieg, A., ‘The graded ring of Hermitian modular forms of degree 2 over $ \mathbb{Q} ( \sqrt{- 2} )$ ’, J. Number Theory 107 (2004) 241–265.CrossRefGoogle Scholar
Dabholkar, A., Murthy, S. and Zagier, D., ‘Quantum black holes, wall crossing, and mock modular forms’, Preprint, 2012, arXiv:hep-th/1208.4074.Google Scholar
Freitag, E. and Hermann, C., ‘Some modular varieties of low dimension’, Adv. Math. 152 (2000) 203–287.CrossRefGoogle Scholar
Gehre, D. and Krieg, A., ‘Quaternionic theta constants’, Arch. Math. 94 (2010) 59–66.CrossRefGoogle Scholar
Gritsenko, V., ‘Arithmetical lifting and its applications’, Séminaire de théorie des nombres de Paris 1992–93 (ed. Sinnou, D.; Cambridge University Press, Cambridge, 1995) 103–126.Google Scholar
Gritsenko, V. and Nikulin, V., ‘Automorphic forms and Lorentzian Kac–Moody algebras II’, Internat. J. Math. 9 (1998) 201–275.CrossRefGoogle Scholar
Hart, W. B. et al. , ‘Fast library for number theory 1.5.2’, 2010, http://www.flintlib.org.Google Scholar
Helgason, S., ‘Differential geometry’, Lie groups, and symmetric spaces, Pure and Applied Mathematics 80 (Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1978).Google Scholar
Harvey, J. and Moore, G., ‘On the algebras of BPS states’, Comm. Math. Phys. 197 (1998) 489–519.CrossRefGoogle Scholar
Heim, B. and Murase, A., ‘Borcherds lifts on ${\mathrm{Sp} }_{2} ( \mathbb{Z} )$ ’, Geometry and analysis of automorphic forms of several variables, Series on Number Theory and Its Applications 7 (World Scientific, Hackensack, NJ, 2012) 56–76.Google Scholar
Jatkar, D. and Sen, A., ‘Dyon spectrum in CHL models’, J. High Energy Phys. 04 (2006) 018.CrossRefGoogle Scholar
Kontsevich, M., ‘Product formulas for modular forms on $\mathrm{O} (2, n)$ (after R. Borcherds)’, Astérisque (1997) no. 245, Exp. No. 821, 3, 41–56, Séminaire Bourbaki, Vol. 1996/97.Google Scholar
Krieg, A., ‘The graded ring of quaternionic modular forms of degree 2’, Math. Z. 251 (2005) 929–942.CrossRefGoogle Scholar
Gopala Krishna, K., ‘BKM Lie superalgebra for the ${Z}_{5} $ -orbifolded CHL string’, J. High Energy Phys. 02 (2012) 089.CrossRefGoogle Scholar
Mayer, S., ‘Calculation of Hilbert Borcherds products’, Experiment. Math. 19 (2010) 243–256.CrossRefGoogle Scholar
O’Meara, O., ‘Introduction to quadratic forms’, Classics in Mathematics (Springer, Berlin, 2000) . Reprint of the 1973 edition.Google Scholar
Raum, M., ‘How to implement a modular form’, J. Symb. Comp. 46 (2011) 1336–1354.CrossRefGoogle Scholar
Raum, M., ‘Computing Jacobi forms and linear equivalences of special divisors’, Preprint, 2012, arXiv: 1212.1834.Google Scholar
Raum, M., ‘Homepage’, 2012, http://www.raum-brothers.eu/martin/.Google Scholar
Raum, M., ‘PSage repository on GitHub’, 2012, https://github.com/martinra/psage.git.Google Scholar
Stein, W. et al. , ‘Purple Sage’, 2011, http://purple.sagemath.org/.Google Scholar
Stein, W. et al. , ‘Sage Mathematics Software (Version 5.0.1)’, The Sage Development Team, 2012,http://www.sagemath.org.Google Scholar
Scheithauer, N., ‘Generalized Kac–Moody algebras, automorphic forms and Conway’s group. I’, Adv. Math. 183 (2004) 240–270.CrossRefGoogle Scholar
Scheithauer, N., ‘Generalized Kac–Moody algebras, automorphic forms and Conway’s group. II’, J. reine angew. Math. 625 (2008) 125–154.Google Scholar
Siegel, C., ‘Indefinite quadratische Formen und Funktionentheorie. I’, Math. Ann. 124 (1951) 17–54.CrossRefGoogle Scholar
Skoruppa, N., ‘Jacobi forms of critical weight and Weil representations’, Modular Forms on Schiermonnikoog (eds Edixhoven, B., Geer, G. V. D. and Moonen, B.; Cambridge University Press, Cambridge, 2008).Google Scholar
Torvald, L. et al. , ‘git version control system’, http://git-scm.com, 2012.Google Scholar