Non-vanishing of Taylor coefficients and Poincaré series

Details Non-vanishing of Taylor coefficients and Poincaré series Non-vanishing of Taylor coefficients and Poincaré series

2010-08-30

arxiv.org/abs/1008.5092v3

Mathematics

Number Theory

23 pages. Accepted for publication by the Ramanujan Journal

Scientific paper

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 Poincar\'e 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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