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Modular forms in Pari/GP

Research in the Mathematical Sciences volume 5, Article number: 37 (2018) Cite this article

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Abstract

We give theoretical and practical information on the Pari/GP modular forms package available since the spring of 2018. Thanks to the use of products of two Eisenstein series, this package is the first which can compute Fourier expansions at any cusps, evaluate modular forms near the real axis, evaluate L-functions of noneigenforms, and compute general Petersson scalar products.

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Notes

  1. 1.

    [0, 1[ is a much more sensible notation than [0, 1), and ]0, 1[ than (0, 1) which can mean so many things.

  2. 2.

    A technicality which explains why representing forms as series with this additional variable is awkward: the variable q must have higher priority than t otherwise some of the examples below will fail. A definition which would work in all cases is mfser(f,n) = Ser(mfcoefs(f,n), varhigher("q", ’t));

References

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    Borisov, L., Gunnells, P.: Toric modular forms and nonvanishing of \(L\)-functions. J. Reine Angew. Math. 539, 149–165 (2001)

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    Borisov, L., Gunnells, P.: Toric modular forms of higher weight. J. Reine Angew. Math. 560, 43–64 (2003)

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    Cohen, H., Oesterlé, J.: Dimensions des espaces de formes modulaires. In: Serre, J.-P., Zagier, D.B. (eds.) Modular Functions of One Variable VI. Lecture Notes in Mathematics, vol. 627, pp. 69–78. Springer, Berlin (1977)

  4. 4.

    Cohen, H., Strömberg, F.: Modular Forms: A Classical Approach. Graduate Studies in Mathematics, vol. 179. American Mathematical Society, Providence (2017)

  5. 5.

    Collins, D.: Numerical computation of Petersson inner products and \(q\)-expansions. arXiv:802.09740 [math]

  6. 6.

    Nelson, P.: Evaluating modular forms on Shimura curves. Math. Comput. 84, 2471–2503 (2015)

  7. 7.

    Skoruppa, N., Zagier, D.: Jacobi forms and a certain space of modular forms. Invent. Math. 94, 113–146 (1988)

  8. 8.

    The PARI Group, PARI/GP version 2.11.0, University of Bordeaux (2018). http://pari.math.u-bordeaux.fr/

  9. 9.

    Weisinger, J.: Some results on classical Eisenstein series and modular forms over function fields. Ph.D. thesis, Harvard University (1977)

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Acknowledgements

We would like to thank B. Allombert, J. Bober, A. Booker, M. Lee, and B. Perrin-Riou for very helpful discussions and help in algorithms and programming, F. Brunault and M. Neururer for Theorem 7, as well as K. Khuri-Makdisi, N. Mascot, N. Billerey and E. Royer. Last but not least, we thank Don Zagier for his continuous input on this package and on Pari/GP in general. In addition, note that a much smaller program written 30 years ago by Don, N. Skoruppa, and the second author can be considered as an ancestor to the present package.

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Affiliations

  1. CNRS, INRIA, IMB, UMR 5251, Univ. Bordeaux, 33400, Talence, France
    • Karim Belabas
    •  & Henri Cohen
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Correspondence to Henri Cohen.

Additional information

Dedicated to Don Zagier for his 65th birthday.

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Belabas, K., Cohen, H. Modular forms in Pari/GP. Res Math Sci 5, 37 (2018) doi:10.1007/s40687-018-0155-z

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