Non-vanishing of Jacobi Poincar\'e series
Abstract
We prove that under suitable conditions, the Jacobi Poincar\'e series of exponential type of integer weight and matrix index does not vanish identically. For classical Jacobi forms, we construct a basis consisting of the "first" few Poincar\'e series and also give conditions both dependent and independent of the weight, which ensures non-vanishing of classical Jacobi Poincar\'e series. Equality of certain Kloosterman-type sums is proved. Also, a result on the non-vanishing of Jacobi Poincar\'e series is obtained when an odd prime divides the index.
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