Non-vanishing of Taylor coefficients and Poincaré series

TY - JOUR

T1 - Non-vanishing of Taylor coefficients and Poincaré series

AU - O'Sullivan, C.

AU - Risager, Morten S.

PY - 2013/1/1

Y1 - 2013/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84872488395&partnerID=8YFLogxK

U2 - 10.1007/s11139-012-9374-x

DO - 10.1007/s11139-012-9374-x

M3 - Journal article

AN - SCOPUS:84872488395

SN - 1382-4090

VL - 30

SP - 67

EP - 100

JO - Ramanujan Journal

JF - Ramanujan Journal

IS - 1

ER -