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Computing Jacobi forms

Part of: Number theory Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  26 August 2016

Nathan C. Ryan ,
Nicolás Sirolli ,
Nils-Peter Skoruppa  and
Gonzalo Tornaría
Nathan C. Ryan
Affiliation:
Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17387, USA email nathan.ryan@bucknell.edu
Nicolás Sirolli
Affiliation:
Universidad de la República, Montevideo, Uruguay email nsirolli@dm.uba.ar Current address:Departamento de Matemática, Ciudad Universitaria, Pabellón I, (C1428EGA) Buenos Aires, Argentina
Nils-Peter Skoruppa
Affiliation:
Universität Siegen, Department Mathematik, 57068 Siegen, Germany email nils.skoruppa@uni-siegen.de
Gonzalo Tornaría
Affiliation:
Centro de Matemática, Facultad de Ciencias, Iguá 4225, (11400) Montevideo, Uruguay email tornaria@cmat.edu.uy

Abstract

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We describe an implementation for computing holomorphic and skew-holomorphic Jacobi forms of integral weight and scalar index on the full modular group. This implementation is based on formulas derived by one of the authors which express Jacobi forms in terms of modular symbols of elliptic modular forms. Since this method allows a Jacobi eigenform to be generated directly from a given modular eigensymbol without reference to the whole ambient space of Jacobi forms, it makes it possible to compute Jacobi Hecke eigenforms of large index. We illustrate our method with several examples.

MSC classification

Primary: 11F50: Jacobi forms 11-04: Explicit machine computation and programs (not the theory of computation or programming)
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 19 , Special Issue A: Algorithmic Number Theory Symposium XII , 2016 , pp. 205 - 219
Copyright
© The Author(s) 2016 

References

Boylan, H., Jacobi forms, finite quadratic modules and Weil representations over number fields , Lecture Notes in Mathematics 2130 (Springer, Cham, 2015) , with a foreword by N.-P. Skoruppa.CrossRefGoogle Scholar
Boylan, H., Skoruppa, N.-P. and Zhou, H., ‘The arithmetic theory of skew-holomorphic Jacobi forms’, Preprint, 2016.Google Scholar
Eichler, M. and Zagier, D., The theory of Jacobi forms , Progress in Mathematics 5 (Birkhäuser, Boston, 1985).Google Scholar
Gross, B., Kohnen, W. and Zagier, D., ‘Heegner points and derivatives of L-series. II’, Math. Ann. 278 (1987) no. 1–4, 497–562.CrossRefGoogle Scholar
Gritsenko, V., Skoruppa, N. and Zagier, D., Theta blocks, in preparation.Google Scholar
The LMFDB Collaboration, ‘The L-functions and modular forms database’, 2015, http://beta.lmfdb.org (accessed 20 December 2015).Google Scholar
Pacetti, A. and Tornaría, G., ‘Computing central values of twisted L-series: the case of composite levels’, Exp. Math. 17 (2008) no. 4, 459–471.CrossRefGoogle Scholar
Ryan, N., Sirolli, N., Skoruppa, N.-P. and Tornaría, G., ‘Jacobi modular forms data’, 2016,http://www.unix.bucknell.edu/∼ncr006/jmf-data/index.html (accessed 10 February 2016).Google Scholar
Skoruppa, N.-P., ‘Binary quadratic forms and the Fourier coefficients of elliptic and Jacobi modular forms’, J. reine angew. Math. 411 (1990) 66–95.Google Scholar
Skoruppa, N.-P., ‘Developments in the theory of Jacobi forms’, Automorphic functions and their applications (Khabarovsk, 1988) (Institute for Applied Mathematics of the USSR Academy of Sciences, Khabarovsk, 1990) 167–185.Google Scholar
Skoruppa, N.-P., ‘Explicit formulas for the Fourier coefficients of Jacobi and elliptic modular forms’, Invent. Math. 102 (1990) no. 3, 501–520.CrossRefGoogle Scholar
Skoruppa, N.-P., ‘Heegner cycles, modular forms and Jacobi forms’, Sém. Théor. Nombres Bordeaux (2) 3 (1991) no. 1, 93–116.Google Scholar
Skoruppa, N.-P., ‘Computations of Siegel modular forms of genus two’, Math. Comp. 58 (1992) no. 197, 381–398.Google Scholar
Skoruppa, N.-P., ‘A remark on Jacobi forms of lattice index’, Waseda Spring Number Theory Conference 2011 (2011).Google Scholar
Skoruppa, N.-P. and Zagier, D., ‘Jacobi forms and a certain space of modular forms’, Invent. Math. 94 (1988) no. 1, 113–146.CrossRefGoogle Scholar
Stein, W., Modular forms, a computational approach , Graduate Studies in Mathematics 79 (American Mathematical Society, Providence, RI, 2007) , with an appendix by Paul E. Gunnells.CrossRefGoogle Scholar
Stein, W. A. et al. , Sage Mathematics Software (Version 6.7), The Sage Development Team, 2015, http://www.sagemath.org.Google Scholar