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Non-vanishing of Taylor coefficients and Poincaré series

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Abstract

We prove recursive formulas for the Taylor coefficients of cusp forms, such as Ramanujan’s Delta function, at points in the upper half-plane. This allows us to show the non-vanishing of all Taylor coefficients of Delta at CM points of small discriminant as well as the non-vanishing of certain Poincaré series. At a “generic” point, all Taylor coefficients are shown to be non-zero. Some conjectures on the Taylor coefficients of Delta at CM points are stated.

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Authors and Affiliations

  1. Department of Mathematics, The CUNY Graduate Center, New York, NY, 10016-4309, USA

    Cormac O’Sullivan

  2. Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen ∅, Denmark

    Morten S. Risager

Authors
  1. Cormac O’Sullivan
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  2. Morten S. Risager
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Correspondence to Cormac O’Sullivan.

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The second author was supported by a grant from The Danish Natural Science Research Council.

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O’Sullivan, C., Risager, M.S. Non-vanishing of Taylor coefficients and Poincaré series. Ramanujan J 30, 67–100 (2013). https://doi.org/10.1007/s11139-012-9374-x

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