Poincaré and Eisenstein series for Jacobi forms of lattice index

Abstract

Poincaré and Eisenstein series are building blocks for every type of automorphic forms. We define Poincaré series for Jacobi forms of lattice index and show that they reproduce Fourier coefficients of cusp forms under the Petersson scalar product. We compute the Fourier expansions of Poincaré and Eisenstein series and give an explicit formula for the Fourier coefficients of the trivial Eisenstein series in terms of values of Dirichlet L-functions at negative integers. For even weight and fixed index, finite linear combinations of Fourier coefficients of non-trivial Eisenstein series are equal to finite linear combinations of Fourier coefficients of the trivial one. This result is used to obtain formulas for the Fourier coefficients of Eisenstein series associated with isotropic elements of small order.