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Forum Math. 11 (1999), 483±502 ( de Gruyter 1999  On some Poincare-series on hyperbolic space Roland Matthes (Communicated by Peter Sarnak)   Abstract. We relate theta-lifts of real-analytic Poincare-series to hyperbolic distance Poincareseries on n-dimensional hyperbolic space averaged over Heegner points and hyperplanes, respectively. As a consequence we can prove at least for the three dimensional case a generalization of a formula of Maass for the Fourier-coe½cients of theta-lifted cusp forms. 1991 Mathematics Subject Classi®cation: 11F. 1 Introduction Let H n denote hyperbolic n-space with dPY Q the hyperbolic distance between the points P and Q. Introduce dPY Q 2 coshdPY Q then it is easily seen (see e.g. [5]) that dPY Q nÀ2 i0 ui P À ui Q 2 v 2 P v 2 Q vPvQ where u0 Y F F F Y unÀ2 Y v are the coordinates in the upper half-space model, see section 2. From the de®nition follows that d is a point pair invariant, i.e. for g from the isometry group of H n we have dgPY Q dPY gÀ1 QX Let q be a discrete co®nite subgroup of Iso H n , of orientation preserving  isometries of H
Forum Mathematicum – de Gruyter
Published: Jun 1, 1999
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