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Computing Power Series Expansions of Modular Forms

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Computations with Modular Forms

Part of the book series: Contributions in Mathematical and Computational Sciences ((CMCS,volume 6))

Abstract

We exhibit a method to numerically compute power series expansions of modular forms on a cocompact Fuchsian group, using the explicit computation of a fundamental domain and linear algebra. As applications, we compute Shimura curve parametrizations of elliptic curves over a totally real field, including the image of CM points, and equations for Shimura curves.

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Acknowledgements

The authors would like to thank Srinath Baba, Valentin Blomer, Noam Elkies, David Gruenewald, Paul Nelson, Kartik Prasanna, Victor Rotger, and Frederik Strömberg for helpful comments on this work. The authors were supported by NSF grant DMS-0901971.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Vermont, 16 Colchester Ave, Burlington, VT, 05401, USA

    John Voight

  2. Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, NH, 03755, USA

    John Voight

  3. Department of Mathematics, The University of Colorado at Boulder, Campus Box 395, Boulder, CO, 80309, USA

    John Willis

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  1. John Voight
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Correspondence to John Voight .

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Editors and Affiliations

  1. Interdisciplinary Center for Scientific Computing, Universität Heidelberg, Heidelberg, Baden-Württemberg, Germany

    Gebhard Böckle

  2. UR Mathématiques, Faculté des Sciences, de la Technologie et de la Communication, Luxembourg, Luxembourg

    Gabor Wiese

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Voight, J., Willis, J. (2014). Computing Power Series Expansions of Modular Forms. In: Böckle, G., Wiese, G. (eds) Computations with Modular Forms. Contributions in Mathematical and Computational Sciences, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-03847-6_13

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