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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)

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